Chapter 7: Study Of Gas Laws
On a December afternoon in 1783, a crowd filled the gardens of the Tuileries in Paris to watch two men climb into a basket slung beneath a huge silk bag. The bag was filled with hydrogen, the lightest gas then known. One of the two men was Jacques Charles, a French physicist who had spent months working out how to make the gas, how to hold it, and how to keep the balloon from tearing itself apart.
That last problem mattered more than the crowd realised. Charles knew that as the balloon rose, the air pressing on it from outside would grow thinner, and the hydrogen inside would push outward and swell. So he built a valve into the top of the bag. As the balloon climbed, he opened the valve and let some hydrogen escape, and the silk stayed whole.
The flight lasted a little over two hours. Later that same day Charles went up again, alone, and rose higher still into air that was noticeably colder.
Charles had done something more useful than fly. He had treated a gas as something that obeys rules — rules connecting how much room it takes up, how hard it pushes, and how warm it is. Those rules are what this chapter is about.
Diagram
Fig. 7.0-1 — Charles’ Hydrogen Balloon Of 1783 Side view of a large spherical silk balloon in flight above a crowd in a garden, with a small basket hanging below it on a net of ropes; label the silk envelope, the hydrogen inside, the release valve at the top of the balloon, the net, and the basket carrying two people; show a small arrow at the valve marked ‘escaping hydrogen’ and a broken outline around the balloon marked ‘gas expands as the balloon rises’.
7.1 Introduction
You have already met the three states of matter. In a solid the particles are locked close together; in a liquid they are still close but able to slide past one another. A gas is the state in which the pull between the particles has almost vanished and the space between them has become very large.
Because so little holds them together, the particles of a gas are free to move about at random through the whole space available to them. This is what defines the gaseous state.
One consequence follows at once. A gas does not settle at the bottom of its container the way a liquid does, and it does not hold a shape of its own the way a solid does. It spreads until it occupies the whole vessel, and so it takes the shape of that vessel. Fig. 7.1-1 shows the same quantity of a gas in three containers of different shapes.
Diagram
Fig. 7.1-1 — A Gas Takes The Shape Of Its Container Three sealed glass containers side by side, all of the same internal volume — a round-bottomed flask, a tall narrow cylinder and a flat wide box — each shown with widely spaced dots spread evenly right up to the walls; label each container and label the dots as ‘gas particles spread through the whole vessel’.
There is something else about gases that makes them unusually easy to study. Solids and liquids differ sharply from one another in the way they respond to heat or to being squeezed, but gases do not. Under the same conditions of temperature and pressure, all gases behave in the same way, whatever their chemical nature, colour or odour. Hydrogen and chlorine are very different substances, yet as gases they expand and contract alike.
That uniform behaviour is why the rest of this chapter can speak simply of ‘a gas’ rather than naming one. The rules you are about to meet apply to all of them.
7.2 Behaviour And Characteristic Properties Of Gases
A gas is made of molecules, and those molecules are never still. They move in every direction, at every sort of speed, colliding with one another and with the walls of the vessel. Fig. 7.2-1 gives the picture you should carry in your head for the whole of this chapter.
Diagram
Fig. 7.2-1 — The Kinetic Picture Of A Gas A closed rectangular box drawn in outline containing about fifteen small circles widely spaced apart, each with a short arrow through it pointing in a different direction and the arrows drawn to different lengths; show two circles meeting in mid-box and two circles striking the right-hand wall; label ‘gas molecules’, ‘arrows show direction of motion, length shows speed’, ‘collision between molecules’ and ‘collision with the wall’.
The set of ideas built on this picture is called the kinetic molecular theory — the explanation of the behaviour of gases in terms of the constant random motion of their molecules. It accounts for the properties described in the eight subsections that follow, and later in the chapter it will account for the gas laws themselves.
7.2.1 Composition Of Gases
A gas is made up of a very large number of tiny particles, which for most gases are molecules. They travel in all possible directions and at all possible speeds.
These molecules are extremely small compared with the vessel they occupy. Nearly all of the space inside a container of gas is empty space between molecules, not the molecules themselves.
7.2.2 Gases Have Neither A Fixed Volume Nor A Fixed Shape
Between the molecules of a gas the force of attraction is negligible. Nothing pulls one molecule back towards its neighbours, so nothing holds the sample together.
The molecules therefore move freely through the entire space open to them. Their movement is checked only by the walls of the container. A gas consequently fills its vessel completely and takes the shape of that vessel.
7.2.3 Gases Exert Pressure In All Directions
The moving molecules of a gas strike one another and also strike the walls of the container. Each strike delivers a small push to the wall.
Any single push is far too small to notice. But an enormous number of molecules strike every second, and together they produce a steady, measurable force on the wall. This is the origin of the pressure of a gas.
At a given temperature, the same number of molecules strike a given area of the wall in a given time, whichever wall it is. That is why a gas presses equally on the top, the bottom and the sides of its vessel — a gas exerts the same pressure in all directions.
7.2.4 Gases Are Highly Compressible
The spaces between the molecules of a gas are large. When you press on a gas, you are mostly squeezing out empty space rather than squeezing the molecules themselves.
Applying pressure therefore brings the molecules closer together and the volume of the gas falls. Gases can be compressed to a small fraction of their original volume, which solids and liquids cannot.
7.2.5 Gases Are Highly Expansible
Gases work the other way round just as readily. Their volume increases when the pressure on them is lowered, and also when their temperature is raised.
Reduce the pressure on an enclosed gas and its molecules move further apart. The spaces between them grow, and so the volume of the gas grows.
Heat an enclosed gas instead and the energy of motion of its molecules increases. They move faster and travel further apart from one another, and again the volume increases.
7.2.6 Gases Have Low Density
Density is mass per unit volume. In a gas, the number of molecules packed into each cubic centimetre is very small compared with a solid or a liquid, because the molecules are so widely spaced.
The same volume of a gas therefore holds far less matter than the same volume of a solid or a liquid. This is why gases have very low densities.
7.2.7 Gases Mix By Diffusion
Because the spaces in a gas are so large, another gas can move straight into them. When two gases are brought into contact, their molecules work their way among one another until the mixture is uniform throughout.
Diffusion in gases is quick and complete. If a jar of chlorine is opened in a large laboratory — something done only inside a fume cupboard, since chlorine is poisonous — its sharp smell reaches every corner of the room within seconds.
What is striking is that chlorine is heavier than air, yet it does not stay near the floor. Its molecules, moving fast in all directions, collide with air molecules and are knocked into fresh directions. Fig. 7.2-2 shows the two sets of molecules working into each other’s spaces.
Diagram
Fig. 7.2-2 — Diffusion Of One Gas Into Another Three panels in a row inside one long horizontal tube. Left panel: filled circles crowded at the left end, open circles at the right end, a dotted line dividing them, labelled ‘at the start’. Middle panel: the two kinds of circles overlapping across the middle, labelled ‘after a short time’. Right panel: filled and open circles evenly mixed along the whole tube, labelled ‘uniform mixture’. Add a key: filled circle = molecule of gas A, open circle = molecule of gas B.
The collisions continue until the concentration of chlorine molecules is the same everywhere in the room. At that stage the mixing is complete, and it does not reverse.
7.2.8 Gases Can Be Liquefied
Cool a gas and its molecules lose energy of motion. They move more slowly, and the weak attractions between them begin to count for something.
Now apply pressure to the cooled gas. The molecules are pushed closer together, the space between them shrinks, and the number of molecules in each unit of volume rises sharply. Brought close enough and moving slowly enough, they hold together and the gas turns into a liquid. This change is called liquefaction.
7.3 Molecular Motion, Temperature, Pressure And Volume
The kinetic picture is useful only if it can be connected to quantities you can actually measure. Those quantities are temperature, pressure and volume. This section makes the three connections, one at a time.
7.3.1 Kinetic Energy And Absolute Temperature
The particles of any substance are in constant random motion, and anything that moves carries energy because it is moving. That energy is called kinetic energy — the energy a body possesses by virtue of its motion.
Molecules in a gas do not all move at the same speed, so they do not all have the same kinetic energy. What matters is the average taken over the whole sample, and that average is fixed by one thing only:
\[\\text{average kinetic energy of a molecule} \\propto T\]
Here \(T\) is the temperature measured on the absolute scale, in kelvin. The relation says that the hotter the gas, the faster its molecules move, on average.
Follow that relation downwards. As the temperature falls, molecular motion slows. If the temperature could be brought to zero on this scale, the motion would stop altogether. That limiting temperature is called absolute zero — the temperature at which molecular motion is taken to cease completely.
Absolute zero lies at \(-273\\ ^\\circ\\text{C}\). It is the reason a second temperature scale is needed at all, and that scale — the Kelvin scale, also called the absolute scale of temperature — is set out fully in section 7.8. No temperature below \(-273\\ ^\\circ\\text{C}\) is possible.
7.3.2 How Moving Molecules Produce Pressure
You have seen that molecules striking a wall press on it. It is worth being exact about what ‘pressure’ means here, because the wall feels a force but what we measure is a pressure.
Force is the total push delivered to the wall. Spread that push over the area of the wall and you have the pressure of the gas.
Now heat the gas, keeping the volume of the vessel unchanged, as in Fig. 7.3-1. Three things change together, and each one adds to the pressure.
Diagram
Fig. 7.3-1 — Heating A Gas At Constant Volume Two identical rigid sealed flasks side by side, each fitted with a pressure gauge at the top and each containing the same number of small circles. Left flask: short arrows on the circles, few contact points marked on the walls, gauge needle low, labelled ‘cool gas — slow molecules, low pressure’. Right flask: long arrows on the circles, many contact points marked on the walls, gauge needle high, labelled ‘hot gas — fast molecules, high pressure’. Label both flasks ‘volume unchanged’.
First, the average kinetic energy of the molecules rises, so their average speed rises. Second, each molecule now arrives at the wall with greater momentum, momentum being the product of the mass of a body and its velocity. A faster molecule delivers a harder blow.
Third, moving faster, the molecules travel between the walls more often, so the rate of collision goes up. More blows, and each one harder, mean a greater force on the walls — and therefore a greater pressure.
7.3.3 How Molecular Motion Changes Volume
The walls of the vessel in Fig. 7.3-1 were rigid, so the gas had no choice but to press harder. Suppose instead that one wall can move, as the piston in Fig. 7.3-2 can.
Diagram
Fig. 7.3-2 — Heating A Gas Behind A Movable Piston Two horizontal cylinders drawn one above the other, each closed at the left end and fitted with a freely sliding piston, with the same weight resting on top of each piston. Upper cylinder: piston near the left, short arrows on the enclosed circles, labelled ‘cool gas — small volume’. Lower cylinder: same cylinder with a burner beneath it, piston pushed further right, long arrows on the circles spread further apart, labelled ‘hot gas — larger volume’. Mark the piston travel with a dimension arrow labelled ‘increase in volume’.
The heated molecules push against the piston harder than the outside air pushes back. The piston is driven outwards until the two pushes balance again.
The gas has expanded instead of building up pressure. So an increase in molecular motion shows itself either as a rise in pressure or as a rise in volume, depending on whether the vessel can give way.
7.4 The Gas Laws And The Standard Variables
You now have a picture of what a gas does and why. The next step is to turn that picture into rules that can be used to predict numbers.
The physical behaviour of any gas can be described by three quantities, called the standard variables:
- volume, \(V\) - pressure, \(P\) - temperature, \(T\)
All gases are alike in their physical behaviour, expanding and contracting equally under the same conditions. For a given mass of a gas, a change in one or more of these three variables produces a change in the remaining ones. That is the whole subject of this chapter, and the three subsections below fix exactly what each variable means and how it is measured.
7.4.1 Volume And Its Units
The volume of a gas is the space occupied by that gas. Since a gas always fills its vessel completely, the volume of a gas is simply the volume of its container.
In the SI system, volume is measured in cubic metre, \(\\text{m}^3\). A cubic metre is far too large for laboratory work, so smaller units are used: the cubic centimetre (\(\\text{cm}^3\)), the millilitre (\(\\text{mL}\)), the cubic decimetre (\(\\text{dm}^3\)) and the litre (\(\\text{L}\)). Table 7.1 relates them.
Table 7.1 — Units of volume and their relationships
| Relationship | |
|---|---|
| \(1\\ \\text{m}^3\) | \(= 1000\\ \\text{dm}^3 = 1000\\ \\text{L}\) |
| \(1\\ \\text{L}\) | \(= 1000\\ \\text{mL} = 1000\\ \\text{cm}^3\) |
Read the second line carefully, because it hides a useful fact: a millilitre and a cubic centimetre are the same size, and so are a litre and a cubic decimetre. A volume given in \(\\text{cm}^3\) can be written as \(\\text{mL}\) without any arithmetic at all.
7.4.2 Pressure And Its Units
Pressure was defined in section 7.3.2 as the force a gas exerts on unit area of the walls of its container. Here we deal with how it is measured.
Our planet is wrapped in a thick blanket of air called the atmosphere, and that air presses on everything beneath it. The pressure exerted by the air of the atmosphere on the surface of the earth is called atmospheric pressure. It is measured with a barometer, a tube of mercury standing in a mercury reservoir, as in Fig. 7.4-1. The air pushes down on the exposed mercury and holds up a column of mercury in the tube; the height of that column is a direct measure of the pressure.
Diagram
Fig. 7.4-1 — Measuring Atmospheric Pressure With A Mercury Barometer A long vertical glass tube, closed at the top and standing with its open lower end dipping into an open dish of mercury; the tube is filled with mercury up to a level well below the closed top. Label the closed space above the mercury as ‘vacuum’, the mercury in the tube, the mercury in the dish, and downward arrows on the surface of the mercury in the dish labelled ‘atmospheric pressure’. Mark the vertical height from the dish surface to the top of the mercury column with a dimension arrow labelled ‘76 cm at standard pressure’.
At sea level the column stands about 76 cm high, and this height is used to fix the unit called the atmosphere.
The conditions in that definition are not decoration. Mercury expands slightly when warmed and the pull of gravity varies a little from place to place, so a column of a stated height means a definite pressure only when the temperature and the gravity are stated too.
The SI unit of pressure is the pascal, symbol Pa. One pascal is the pressure exerted when a force of 1 newton acts on an area of \(1\\ \\text{m}^2\). Alongside it, chemists still use the atmosphere and the older units taken straight from the barometer — centimetre of mercury (cm Hg), millimetre of mercury (mm Hg), and the torr. Table 7.2 gives the relationships you will need.
Table 7.2 — Units of pressure and their relationships
| Unit | Symbol | Relationship |
|---|---|---|
| Pascal | Pa | SI unit; 1 Pa is a force of 1 newton on an area of \(1\\ \\text{m}^2\) |
| Atmosphere | atm | \(1\\ \\text{atm} = 76.0\\ \\text{cm Hg}\) |
| Centimetre of mercury | cm Hg | \(76.0\\ \\text{cm Hg} = 760\\ \\text{mm Hg}\) |
| Millimetre of mercury | mm Hg | \(760\\ \\text{mm Hg} = 760\\ \\text{torr}\) |
| Torr | torr | \(1\\ \\text{atm} = 760\\ \\text{torr}\) |
Because a gas-law problem may hand you one unit and ask for another, it is worth practising the conversions on their own before the laws arrive.
7.4.3 Temperature And Its Scales
The temperature of a gas is defined as the degree of hotness of that gas.
The scale in everyday use is the Celsius scale, once known as the centigrade scale. Its zero is purely a matter of choice — it was fixed at the melting point of ice, which is convenient but not special. Because the zero is arbitrary, a substance can perfectly well have a temperature below it, and negative Celsius values are ordinary.
The behaviour of gases, however, shows that temperatures below \(-273\\ ^\\circ\\text{C}\) are not possible at all. There is a real floor, and the Celsius scale places its zero nowhere near it. This is what led to the second scale, the Kelvin scale, whose zero is placed at that floor: \(0\\ \\text{K}\) corresponds to \(-273\\ ^\\circ\\text{C}\).
One degree on the Kelvin scale is the same size as one degree on the Celsius scale, so the two scales differ only in where they start counting. Converting between them is therefore a matter of adding or subtracting 273, and section 7.9 works through it. From here onwards, every gas-law calculation in this chapter uses temperature on the Kelvin scale.
7.5 Pressure And Volume Relationship In Gases
Of the three standard variables, two can be studied together if the third is held fixed. Keep the temperature constant and you are left with pressure and volume — and a relationship you can find out for yourself with a syringe.
7.5.1 The Syringe Experiment
Take a 10 mL syringe fitted with a piston. Raise the piston to the 10 mL mark so that a known volume of air is trapped inside, and wrap an adhesive tape over the nozzle to seal it. Fit the wrapped nozzle tightly into a hole bored half way through a rubber stopper, which holds the syringe upright and steady. The arrangement is shown in Fig. 7.5-1.
Diagram
Fig. 7.5-1 — Studying The Pressure–Volume Relationship With A Syringe A syringe standing vertically, nozzle downward, its nozzle wrapped in tape and pushed into a hole bored half way through a rubber stopper resting on the bench; the piston is part way down the barrel with trapped air below it, and small weights are stacked on the flat top of the piston. Label the weights, the piston, the syringe barrel, the trapped air column, the taped nozzle and the rubber stopper. Add a vertical dimension arrow beside the trapped air labelled ‘volume of air’.
Now place some weight on the piston. Pressing down on the piston raises the pressure on the trapped air, and you will see the piston move downward, reducing the volume of the air. Add more weight, gradually, and the piston travels further down and the volume is reduced further.
Then reverse the change. Remove the weights one by one, and you will notice the piston moving upward as the pressure falls, so that the volume of the air increases.
What The Experiment Shows
Two conclusions follow, and each is a two-way statement.
- An increase in pressure at constant temperature causes a decrease in the volume of a gas; conversely, if the volume of a fixed mass of a gas at constant temperature is decreased, the pressure of the gas increases.
- A decrease in pressure at constant temperature causes an increase in the volume of a gas; conversely, if the volume of a fixed mass of a gas at constant temperature is increased, the pressure of the gas decreases.
7.5.2 Boyle’s Law And Its Mathematical Form
Robert Boyle studied this relationship systematically. In 1662 he found that, at constant temperature, the volume of a fixed mass of a dry gas decreased by half when the pressure on it was doubled, and became four times its original volume when the pressure was reduced to one-fourth. He set the behaviour out as a law.
Three conditions are built into that sentence and none of them can be dropped. The mass of gas must be fixed, the temperature must be constant, and the gas must be dry — a moist gas carries water vapour, which contributes a pressure of its own, as section 7.12 explains.
Writing The Law As An Equation
Suppose a gas occupies volume \(V\_1\) when its pressure is \(P\_1\). Inverse proportion means
\[V\_1 \\propto \\frac{1}{P\_1} \\qquad \\text{or} \\qquad V\_1 = \\frac{k}{P\_1}\]
where \(k\) is a constant for that sample at that temperature. Multiplying both sides by \(P\_1\),
\[P\_1V\_1 = k\]
Let the same gas now occupy volume \(V\_2\) at pressure \(P\_2\), the temperature unchanged. By the same argument,
\[P\_2V\_2 = k\]
Both products equal the same constant, so they equal each other:
\[P\_1V\_1 = P\_2V\_2 = k \\qquad \\text{(at constant temperature)}\]
This is called Boyle’s law equation. It is the form you will actually use, because it lets you find any one of the four quantities when the other three are known.
The same result can be stated in words without any proportion at all.
7.5.3 Graphical Verification Of Boyle’s Law
A law of this kind can be tested by plotting it. Three graphs are used, and each shows the same law from a different angle: \(V\) against \(P\), \(V\) against \(\\frac{1}{P}\), and \(PV\) against \(P\).
Volume Plotted Against Pressure
Consider four litres of a dry gas at a pressure of one atmosphere, at constant temperature. Increase the pressure uniformly from one atmosphere to four atmospheres, and the volume of the dry gas is observed to fall steadily, as in Fig. 7.5-2.
Diagram
Fig. 7.5-2 — Variation In Volume Plotted Against Pressure At Constant Temperature Graph with ‘Pressure (atmosphere) →’ on the x-axis marked 1, 2, 3, 4 and ‘V (litres) ↑’ on the y-axis marked 1, 2, 3, 4; a smooth curve falling steeply then flattening, passing through the points (1, 4), (2, 2) and (4, 1) and approaching but never touching either axis. Label the curve ‘hyperbolic curve’ and add the note ‘temperature is constant’ on the plot, with dashed guide lines dropped from the three named points to both axes.
The curve is a hyperbola lying in the first quadrant, exactly what inverse proportion produces. Such a graph, drawn at a constant temperature, is called an isotherm, the word meaning ‘at constant temperature’.
Volume Plotted Against The Reciprocal Of Pressure
The curve is convincing but awkward to read. Plot the volume against \(\\frac{1}{P}\) instead, at constant temperature, and the same data give a straight line passing through the origin, as in Fig. 7.5-3.
Diagram
Fig. 7.5-3 — Variation In Volume Plotted Against 1/P At Constant Temperature Graph with ‘1/P →’ on the x-axis marked 1, 2, 3, 4 and ‘V ↑’ on the y-axis marked 1, 2, 3, 4; a single straight line starting exactly at the origin and rising at a steady slope through the marked points. Add the note ‘temperature is constant’ on the plot.
A straight line through the origin is the signature of direct proportion. So the volume is directly proportional to the inverse of the pressure — which is another way of saying it is inversely proportional to the pressure. Boyle’s law again.
The Product PV Plotted Against Pressure
The third graph tests the second statement of the law. For each pressure, multiply the pressure by the volume measured at it, and plot that product against the pressure. The result, in Fig. 7.5-4, is a straight line parallel to the pressure axis.
Diagram
Fig. 7.5-4 — Variation In PV Plotted Against P At A Constant Temperature Graph with ‘P →’ on the x-axis marked 1 to 5 and ‘PV ↑’ on the y-axis marked 1 to 4; five plotted points all at a height of 4 joined by a horizontal straight line running parallel to the x-axis, the line labelled ‘T’ at its right-hand end to show a fixed temperature.
A horizontal line means the quantity plotted upwards is not changing. The product of pressure and volume of a given mass of a gas stays constant while the temperature stays constant. Table 7.3 gives the figures behind the line.
Table 7.3 — Pressure, volume and their product for a fixed mass of gas at constant temperature
| Pressure (P) | Volume (V) | (P × V) |
|---|---|---|
| 1 atm | 4.0 L | 4 |
| 2 atm | 2.0 L | 4 |
| 4 atm | 1.0 L | 4 |
7.5.4 Boyle’s Law Explained By Molecular Motion
A law that is only measured is a rule without a reason. The kinetic theory supplies the reason.
For a given mass of a gas at a constant temperature, two things are fixed: the number of molecules present, and the average kinetic energy each one possesses. Neither the number of blows available nor the strength of each blow can change.
Now halve the volume. The same number of molecules are confined to half the space, so they have half as far to travel between the walls. In a given time, twice as many molecules strike unit area of the wall — and so the pressure is doubled.
Run it the other way. Double the volume at constant temperature and the same molecules have twice as much room to move about in. The number striking unit area of the wall in a given time falls to one half of its original value, and the pressure falls to half its original value.
So a decrease in volume raises the pressure and an increase in volume lowers it — which is Boyle’s law, arrived at from molecular motion alone.
7.5.5 Significance Of Boyle’s Law
Raising the pressure on a gas squeezes the same mass into a smaller volume. The gas therefore becomes denser. Since density is mass divided by volume, and the volume falls in step with the rise in pressure, it follows that at constant temperature the density of a gas is directly proportional to its pressure.
This has a consequence you can feel. Atmospheric pressure is low at high altitudes, so the air there is less dense. A given lungful contains a lesser quantity of oxygen than the same lungful at sea level. That is why mountaineers have to carry oxygen cylinders with them.
7.5.6 Boyle’s Law In Everyday Life
Breathing Mechanism
As the lungs expand, their volume increases. By Boyle’s law the pressure inside the lungs falls, and air flows in from outside, where the pressure is higher. That is inhalation.
When the lungs contract, their volume decreases and the pressure inside rises, pushing the air out again. That is exhalation.
Working Of Syringes
When the plunger of a syringe is pulled back, the volume inside the tube increases and the pressure inside falls. External fluid is pushed in by the higher outside pressure. When the plunger is pushed in, the volume decreases and the pressure rises, forcing the fluid out.
Scuba Diving
Underwater, the pressure increases with depth, because water is much denser than air. For every 10 metres of depth, the pressure increases by nearly one atmosphere.
As a diver ascends towards the surface, the pressure falls, and the air held in body cavities expands in accordance with Boyle’s law. For this reason divers must ascend slowly, allowing the body to adjust to the changing pressure and preventing overexpansion of the lungs.
A Case That Only Looks Like An Exception
Blow air into a balloon and both the volume and the pressure inside it increase together. Boyle’s law seems to be broken.
It is not. The law holds for a definite mass of gas, and here the mass is not definite — more air is being added with every breath. Boyle’s law says nothing about what happens when you change the amount of gas.
Working With Boyle’s Law
In every calculation below, the temperature is constant and the mass of gas is fixed, so \(P\_1V\_1 = P\_2V\_2\) applies. Both pressures must be in the same unit and both volumes in the same unit; the unit itself does not matter, because it cancels.
7.6 Temperature And Volume Relationship In Gases
Leave an inflated balloon in bright sunshine and after a while it bursts. Nothing has been added to it. The air already inside has expanded with the rise in temperature until the rubber can hold it no longer.
In general, an increase in temperature at constant pressure causes an increase in volume, and a decrease in temperature at constant pressure causes a decrease in volume. This section turns that general statement into a law with a number attached to it.
7.6.1 Charles’ Law And Its Mathematical Form
In 1787, Jacques Charles studied the temperature–volume relationship of gases. His work was further extended by Joseph Gay-Lussac in 1802.
What the measurements showed was a fixed fraction, the same for every gas. A fixed mass of any gas expands or contracts respectively by \(\\frac{1}{273}\) of its volume at \(0\\ ^\\circ\\text{C}\) for each degree Celsius rise or fall in temperature, provided the pressure remains constant.
Building The Mathematical Expression
Let \(V\_0\) be the volume of a fixed mass of a gas at \(0\\ ^\\circ\\text{C}\), and let \(V\) be its volume at temperature \(t\\ ^\\circ\\text{C}\), at constant pressure. Each degree adds \(\\frac{V\_0}{273}\), and there are \(t\) degrees, so according to Charles’ law
\[V = V\_0 + \\frac{V\_0}{273},t \\qquad \\text{(when } P \\text{ is constant)}\]
Take \(V\_0\) out as a common factor and add the terms inside the bracket:
\[V = V\_0\\left(1 + \\frac{t}{273}\\right) = V\_0\\left(\\frac{273 + t}{273}\\right) \\qquad \\dots (1)\]
Now look at what sits inside that last bracket. The quantity \(273 + t\) is exactly the Celsius temperature converted to the Kelvin scale, so write \(T = 273 + t\):
\[V = \\frac{V\_0}{273},T\]
For a given mass of a gas, \(V\_0\) is a fixed volume and 273 is a fixed number, so
\[\\frac{V\_0}{273} = \\text{constant}\]
which leaves
\[V = k \\times T \\qquad \\text{(where } k \\text{ is a constant)}\]
\[\\text{or} \\qquad V \\propto T \\qquad \\text{and} \\qquad \\frac{V}{T} = k\]
The awkward fraction has disappeared, and a much cleaner statement of the same law has taken its place.
Suppose now that a gas occupies \(V\_1\\ \\text{cm}^3\) at temperature \(T\_1\) and \(V\_2\\ \\text{cm}^3\) at temperature \(T\_2\), the pressure unchanged. By Charles’ law,
\[V\_1 \\propto T\_1 \\qquad \\text{or} \\qquad V\_1 = kT\_1 \\qquad \\text{so} \\qquad \\frac{V\_1}{T\_1} = k\]
\[V\_2 \\propto T\_2 \\qquad \\text{or} \\qquad V\_2 = kT\_2 \\qquad \\text{so} \\qquad \\frac{V\_2}{T\_2} = k\]
Both ratios equal the same constant, and therefore
\[\\frac{V\_1}{T\_1} = \\frac{V\_2}{T\_2} = k \\qquad \\text{(at constant pressure)}\]
This is called the Charles’ law equation. Two consequences read straight off it:
- If the absolute temperature is doubled, the volume would be doubled.
- If the absolute temperature is reduced by half, the volume would also be reduced by half.
7.6.2 Graphical Representation Of Charles’ Law
Plot the measured volume of a fixed mass of gas against its temperature in degrees Celsius, keeping the pressure constant, and the points fall on a straight line, as in Fig. 7.6-1.
Diagram
Fig. 7.6-1 — Graph Showing The Volume–Temperature Relationship In A Gas At Constant Pressure Graph with ‘Temperature (°C) →’ on the x-axis marked at −273 °C, −200 °C, −100 °C, 0 °C, 100 °C, 200 °C, 300 °C and 400 °C, and ‘Volume (mL) ↑’ on the y-axis marked 10.0 to 80.0 in steps of 10.0. A solid straight line rises steadily from left to right across the measured region to the right of 0 °C, with a few plotted points on it; the line is continued to the left of 0 °C as a broken line that meets the x-axis exactly at −273 °C. Label the solid part ‘measured values’ and the broken part ‘line extended backwards’.
A straight line is obtained, which proves that volume is directly proportional to temperature, provided the pressure remains constant. A graph drawn at a constant pressure is given the general term isobar, the word meaning ‘at constant pressure’.
The experiment cannot be carried down to very low temperatures, because every gas turns to a liquid long before then. But the straight line can be continued backwards on paper, and Fig. 7.6-2 shows what happens when this is done for several different constant pressures.
Diagram
Fig. 7.6-2 — Isobars Extended Backwards To Show The Absolute Zero Point Graph with ‘Temperature →’ on the x-axis, marked at −273 °C and at 100 °C, and ‘Volume ↑’ on the y-axis. Three solid straight lines of different slopes rise to the right, labelled P₁ (steepest), P₂ and P₃ (least steep); each is continued to the left as a broken line, and all three broken lines meet the x-axis at the single point marked −273 °C. Label that point ‘absolute zero’.
Each pressure gives a line of its own slope, yet all of them, extended backwards, cut the temperature axis at the same point: \(-273\\ ^\\circ\\text{C}\). Where that point leads is the subject of section 7.7.
7.6.3 Charles’ Law Explained By Molecular Motion
The kinetic theory explains this law as readily as it explained the last one. The average kinetic energy of the gas molecules is directly proportional to the absolute temperature.
Raise the temperature of a gas and its molecules move faster. They strike the walls of the container more frequently and more vigorously than before, and the gas presses harder.
But here the pressure is being kept constant. The only way the pressure can stay where it was, while each blow is harder and the blows come more often, is for the wall to give way — the gas expands until the blows are spread over a larger area and arrive less often. The volume increases in proportion.
Hence, at constant pressure, the volume of a given mass of a gas is directly proportional to its absolute temperature, which is Charles’ law.
7.6.4 Significance Of Charles’ Law
Since the volume of a given mass of a gas is directly proportional to its temperature, the same mass spreads over a larger volume as it is heated. Density is mass divided by volume, so the density of a gas decreases with an increase in temperature.
This is the reason why hot air is filled into the balloons used for meteorological purposes. The heated air inside is less dense than the cooler air outside, and the balloon rises through it, carrying instruments up into the atmosphere.
7.6.5 Charles’ Law In Everyday Life
Food Preservation Techniques
Canning and freezing methods of preserving food for extended periods rely on Charles’ law. Lowering the temperature reduces the volume of the gases inside the sealed container, which reduces the pressure inside it.
The result is a partial vacuum sealed into the container. That seal keeps out fresh air, so oxidation and bacterial growth are minimised and the food stays fresh for longer without spoiling.
Hot Air Balloons
In a hot air balloon, heating the air inside the envelope causes it to expand and become less dense than the surrounding cooler air. The difference in density lifts the balloon, which is Charles’ law in action on a large scale.
Tyre Pressure
When tyres heat up through friction with the road, the air inside them tends to expand. A tyre has a nearly fixed volume, so the expansion shows up instead as a rise in the internal pressure.
Tyre pressure readings therefore vary with temperature, and a reading taken after a long drive will be higher than the true one. This is why tyre pressure should always be measured under consistent conditions, on cold tyres.
Working With Charles’ Law
Every calculation below keeps the pressure and the mass of gas constant, so \(\\dfrac{V\_1}{T\_1} = \\dfrac{V\_2}{T\_2}\) applies. Convert each temperature to kelvin before substituting, and keep both volumes in the same unit.
7.7 Absolute Zero
The broken lines in Fig. 7.6-2 all met the temperature axis at \(-273\\ ^\\circ\\text{C}\). That meeting point can also be reached by arithmetic, without drawing anything at all.
Take equation (1) from section 7.6.1:
\[V = V\_0\\left(\\frac{273 + t}{273}\\right)\]
Substitute \(t = -273\), which is the temperature at which the lines met:
\[V \\text{ at } -273\\ ^\\circ\\text{C} = V\_0\\left(\\frac{273 - 273}{273}\\right) = 0\]
So Charles’ law, followed strictly, predicts that the volume of a gas would be reduced to zero at \(-273\\ ^\\circ\\text{C}\). Lord Kelvin drew attention to exactly this: if the temperature of an enclosed gas at \(0\\ ^\\circ\\text{C}\) is lowered to \(-273\\ ^\\circ\\text{C}\), its volume becomes zero.
That result cannot be taken literally. A gas is one of the states of matter, and matter must have some definite mass and occupy some definite volume. A sample of matter cannot simply shrink out of existence.
What saves the situation is that no gas ever gets that far. All gases liquefy or solidify well before reaching this temperature, and once the substance is no longer a gas, Charles’ law no longer applies to it. The prediction is never put to the test.
Nevertheless, the temperature \(-273\\ ^\\circ\\text{C}\) keeps its importance. It is called absolute zero, and theoretically it is the lowest temperature that can ever be reached. At this temperature all molecular motion ceases. Practically speaking, however, it is impossible to attain.
7.8 The Absolute Or Kelvin Scale Of Temperature
Once absolute zero is known, a better temperature scale can be built. Start counting from the coldest temperature there is, and keep the degrees the same size as the familiar ones.
The unit of this scale is the kelvin, symbol K. Because its degrees are the same size as Celsius degrees, the two scales run side by side with a fixed gap of 273 between them, as Fig. 7.8-1 shows.
Diagram
Fig. 7.8-1 — Celsius And Kelvin Thermometers Compared Two thermometer bulbs and stems drawn side by side and aligned at the same heights, the left one headed ‘CELSIUS °C’ and the right one headed ‘KELVIN K’. On the left stem mark 100 at the upper fixed point and 0 at the lower fixed point; on the right stem mark 373 and 273 at the same two heights. Label the upper level ‘UPPER FIXED POINT — STEAM POINT’ and the lower level ‘LOWER FIXED POINT — ICE POINT’. Between the two marks on the left, draw a double-headed vertical arrow labelled ‘100 °C’; on the right, the matching arrow labelled ‘100 K’. Add one intermediate level marked 20 °C on the left and 293 K on the right, joined by a dotted horizontal line.
Notice what the figure is saying. The ice point and the steam point are the same two physical events on both scales — ice melting, water boiling. Only the numbers attached to them differ, and the interval between them is one hundred divisions on each scale.
Three Things To Remember About This Scale
- No degree sign is written for absolute temperature values. Water boils at \(100\\ ^\\circ\\text{C}\), but at \(373\\ \\text{K}\) — never \(373\\ ^\\circ\\text{K}\).
- The size of 1 degree on the Kelvin scale is the same as the size of 1 degree on the Celsius scale. A rise of 5 kelvin and a rise of \(5\\ ^\\circ\\text{C}\) are the same rise.
- The real advantage of the Kelvin scale is that it makes the application and use of the gas laws simple. Even more significantly, all values on the Kelvin scale are positive.
That last point matters more than it may seem. Charles’ law says the volume is directly proportional to the temperature, and a volume can never be negative. On the Celsius scale, temperatures below zero are common, and the proportion would be meaningless. On the Kelvin scale there are no negative temperatures at all, so the proportion always makes sense.
7.9 Conversion Between The Celsius And Kelvin Scales
Since the two scales differ only in where they start counting, moving between them is a single addition or subtraction. Every gas-law problem in this chapter begins with this step, so it is worth doing until it becomes automatic.
The value on the Celsius scale can be converted to the Kelvin scale by adding 273 to it.
\[\\text{Kelvin (K)} = ^\\circ\\text{C} + 273\]
\[0\\ ^\\circ\\text{C} = 0 + 273 = 273\\ \\text{K}\] \[20\\ ^\\circ\\text{C} = 20 + 273 = 293\\ \\text{K}\] \[27\\ ^\\circ\\text{C} = 27 + 273 = 300\\ \\text{K}\] \[-273\\ ^\\circ\\text{C} = -273 + 273 = 0\\ \\text{K}\]
The last line is worth pausing over. It is the definition of the scale doing its work: absolute zero on the Celsius scale becomes the zero of the Kelvin scale.
Kelvin scale values can be converted to degree Celsius values by subtracting 273 from them.
\[0\\ \\text{K} = 0 - 273 = -273\\ ^\\circ\\text{C}\] \[13\\ \\text{K} = 13 - 273 = -260\\ ^\\circ\\text{C}\] \[273\\ \\text{K} = 273 - 273 = 0\\ ^\\circ\\text{C} \\qquad \\text{(freezing point of water, or ice point)}\] \[373\\ \\text{K} = 373 - 273 = 100\\ ^\\circ\\text{C} \\qquad \\text{(boiling point of water)}\]
Table 7.4 collects the values you will meet most often.
Table 7.4 — Some temperatures on both scales
| Celsius scale | Kelvin scale | What it is |
|---|---|---|
| \(-273\\ ^\\circ\\text{C}\) | \(0\\ \\text{K}\) | absolute zero |
| \(0\\ ^\\circ\\text{C}\) | \(273\\ \\text{K}\) | ice point |
| \(27\\ ^\\circ\\text{C}\) | \(300\\ \\text{K}\) | a warm room |
| \(100\\ ^\\circ\\text{C}\) | \(373\\ \\text{K}\) | steam point |
A quick check will save you many errors: a Kelvin value is always the larger of the pair, and it is never negative.
7.10 The Gas Equation
So far, one variable has been held still while the other two were studied. Real changes are rarely so tidy. If both the pressure and the temperature of a fixed mass of a gas are varied together, a combination of Boyle’s and Charles’ laws gives a single fixed relation among its volume, pressure and temperature.
Start from the two laws already established. According to Boyle’s law,
\[V \\propto \\frac{1}{P}\]
and according to Charles’ law,
\[V \\propto T\]
Now combine the two. The volume of a given mass of a gas varies inversely with pressure and directly with absolute temperature, so both factors act on it at once:
\[V \\propto \\frac{1}{P} \\times T\]
\[\\text{or} \\qquad V = \\frac{T}{P} \\times \\text{constant}\]
Multiply both sides by \(P\) and divide by \(T\), and the constant is left standing alone:
\[\\frac{PV}{T} = \\text{constant}\]
Accordingly, if the volume of a given mass of a gas changes from \(V\_1\) to \(V\_2\), its pressure from \(P\_1\) to \(P\_2\), and its temperature from \(T\_1\) to \(T\_2\), then both sets of values give the same constant:
\[\\frac{P\_1V\_1}{T\_1} = \\frac{P\_2V\_2}{T\_2}\]
This expression is called the gas equation. It is used for calculating changes in gas volume when both the pressure and the temperature change.
The Two Laws Are Still Inside It
The gas equation does not replace Boyle’s law and Charles’ law; it contains them. Hold the temperature constant, so that \(T\_1 = T\_2\), and the temperatures cancel, leaving \(P\_1V\_1 = P\_2V\_2\) — Boyle’s law. Hold the pressure constant instead, so that \(P\_1 = P\_2\), and the pressures cancel, leaving \(\\frac{V\_1}{T\_1} = \\frac{V\_2}{T\_2}\) — Charles’ law. FC 7.1 sets out the choice.
Flowchart
FC 7.1 — Choosing The Right Relation For A Gas Problem List what changes → is the temperature constant? → yes: use P₁V₁ = P₂V₂ → no: is the pressure constant? → yes: use V₁/T₁ = V₂/T₂ → no: both change, use P₁V₁/T₁ = P₂V₂/T₂ → in every case, convert all temperatures to kelvin and match the units of the two pressures and of the two volumes before substituting
There is a further point hidden in the equation. Increasing the pressure on an enclosed gas decreases its volume, whereas increasing its temperature increases its volume. The two effects pull in opposite directions. By keeping a balance between an increase in pressure and an increase in temperature, it is therefore possible to keep the volume constant, as Example 7.14 shows.
Working With The Gas Equation
Set out the initial values on one side and the final values on the other before substituting anything. Temperatures go in kelvin; the two pressures share one unit and the two volumes share another.
7.11 Standard Temperature And Pressure
Everything in this chapter has shown that the volume of a gas changes remarkably with a change in temperature and in pressure. A volume quoted on its own therefore says almost nothing.
Suppose a laboratory in one city reports that a reaction gave \(50\\ \\text{cm}^3\) of a gas, and a laboratory in another city reports \(54\\ \\text{cm}^3\) from the same reaction. Until the temperature and pressure of each measurement are known, there is no way to tell whether the two results agree or disagree.
It becomes necessary, then, to choose standard values of temperature and pressure to which gas volumes can be referred. Any measurement can be converted to those conditions using the gas equation, and the converted volumes can then be compared fairly.
The standard values chosen are \(0\\ ^\\circ\\text{C}\), or \(273\\ \\text{K}\), for temperature, and 1 atmospheric unit (atm), or \(760\\ \\text{mm Hg}\), for pressure. These agreed values are known as standard temperature and pressure, written S.T.P.
Table 7.5 gives the standard values in every unit you are likely to meet them in.
Table 7.5 — Standard temperature and pressure
| Quantity | Standard value |
|---|---|
| Standard temperature | \(0\\ ^\\circ\\text{C} = 273\\ \\text{K}\) |
| Standard pressure | \(760\\ \\text{mm Hg} = 76\\ \\text{cm Hg} = 1\\ \\text{atm}\) |
Any change in pressure or temperature is likely to change the volume of a gas. Thus, while expressing the specific volume of a gas, it is necessary to specify both its pressure and its temperature — either by stating them, or by stating that the volume has been converted to S.T.P.
7.12 The Effect Of Moisture On Pressure
One complication remains, and it appears the moment a gas is prepared in the laboratory rather than read about on paper.
Certain gases, such as nitrogen and hydrogen, are insoluble or only slightly soluble in water. A gas of this kind can be collected by filling a jar with water, inverting it over a trough of water, and leading the gas up into it so that the gas pushes the water down and takes its place. This is called collection by the downward displacement of water, and the arrangement is shown in Fig. 7.12-1.
Diagram
Fig. 7.12-1 — Collection Of A Gas Over Water A trough of water with an inverted gas jar standing in it, supported so that its mouth stays below the water surface; a delivery tube from the left dips under the water and turns upward into the mouth of the jar, with bubbles drawn rising from its end into the jar. The upper part of the jar holds the collected gas and the lower part still holds water. Label the delivery tube, the bubbles of gas, the water in the trough, ‘gas + water vapour’ in the space at the top of the jar, and a downward arrow on the water surface of the trough labelled ‘atmospheric pressure’.
The method is simple, but it has a consequence. When a gas is collected over water it is moist — it carries water vapour mixed in with it, because water evaporates into any space above it.
So the jar does not contain one gas. It contains two, and each of them presses on the walls independently of the other. The pressure that one gas of a mixture would exert if it alone filled the containing vessel, at the same temperature, is called its partial pressure.
The total pressure exerted by the moist gas is therefore equal to the sum of the partial pressure of the dry gas and the pressure exerted by the water vapour. The partial pressure of water vapour has a name of its own: it is called the aqueous tension.
Written as equations,
\[P\_{\\text{total}} = P\_{\\text{gas}} + P\_{\\text{water vapour}}\]
\[P\_{\\text{gas}} = P\_{\\text{total}} - P\_{\\text{water vapour}}\]
\[\\text{actual pressure of gas} = \\text{total pressure} - \\text{aqueous tension}\]
Thus, to obtain the actual pressure of the gas you have collected, the pressure which is due to the moisture in it has to be subtracted. Only that corrected pressure may be put into the gas equation, because the gas laws as stated apply to a dry gas.
Aqueous tension depends only on the temperature, and its value at a given temperature is looked up rather than calculated. A problem that requires it will always supply it.
When The Gas Is Given By Mass
A problem may hand you a mass of gas rather than a volume. It can still be solved with the gas equation, provided you are told what volume one molar mass of a gas occupies at S.T.P.
7.13 Summary
- The gaseous state is the state of matter in which the inter-particle attraction is weak and the inter-particle space is so large that the particles move randomly through the entire available space.
- A gas fills its vessel completely and takes its shape. All gases show uniform behaviour under similar conditions of temperature and pressure, whatever their chemical nature, colour or odour.
- The kinetic molecular theory explains the properties of gases in terms of the constant random motion of their molecules.
- Gases are made of tiny molecules moving in all directions at all speeds; the molecules are negligibly small compared with the volume occupied by the gas.
- Because the attraction between the molecules is negligible, gases have neither a fixed volume nor a fixed shape.
- Molecules colliding with the walls exert pressure, and because equal numbers strike each wall in a given time, gases exert the same pressure in all directions.
- Gases are highly compressible and highly expansible, because the spaces between their molecules are large.
- Gases have very low density, since the number of molecules per unit volume is very small compared with solids and liquids.
- Diffusion is the process of gradual mixing of two substances, kept in contact, by molecular motion. Gases diffuse quickly and completely.
- On cooling and applying pressure, gas molecules slow down and come closer, and the gas liquefies.
- Kinetic energy is the energy a body possesses by virtue of its motion. The average kinetic energy of a molecule is directly proportional to the absolute temperature.
- The force exerted on a unit area of the wall of the vessel is equal to the pressure of the gas. Heating a gas at constant volume raises its pressure; if a wall can move, the gas expands instead.
- The behaviour of a gas under known conditions of pressure, volume and temperature is described by the gas laws. The three standard variables are volume \(V\), pressure \(P\) and temperature \(T\).
- \(1\\ \\text{m}^3 = 1000\\ \\text{dm}^3 = 1000\\ \\text{L}\), and \(1\\ \\text{L} = 1000\\ \\text{mL} = 1000\\ \\text{cm}^3\).
- One atmosphere is the pressure exerted by 76 cm of mercury at \(0\\ ^\\circ\\text{C}\) at standard gravity of \(9.8\\ \\text{m s}^{-2}\); \(1\\ \\text{atm} = 76.0\\ \\text{cm Hg} = 760\\ \\text{mm Hg} = 760\\ \\text{torr}\). The SI unit of pressure is the pascal, the pressure of a force of 1 newton on an area of \(1\\ \\text{m}^2\).
- Boyle’s law: the volume of a given mass of a dry gas is inversely proportional to its pressure at constant temperature. \[P\_1V\_1 = P\_2V\_2\]
- Graphically, \(V\) against \(P\) is a hyperbola called an isotherm; \(V\) against \(\\frac{1}{P}\) is a straight line through the origin; \(PV\) against \(P\) is a straight line parallel to the pressure axis.
- Kinetically, halving the volume at constant temperature doubles the number of molecules striking unit area of the wall in a given time, and so doubles the pressure.
- At constant temperature the density of a gas is directly proportional to its pressure. Air is less dense at high altitudes, which is why mountaineers carry oxygen cylinders.
- Charles’ law: pressure remaining constant, the volume of a given mass of a dry gas increases or decreases by \(\\frac{1}{273}\) of its volume at \(0\\ ^\\circ\\text{C}\) for each \(1\\ ^\\circ\\text{C}\) increase or decrease in temperature. Equivalently, the volume is directly proportional to the absolute temperature at constant pressure. \[\\frac{V\_1}{T\_1} = \\frac{V\_2}{T\_2}\]
- A volume–temperature graph at constant pressure is a straight line called an isobar. Extended backwards, isobars drawn at different pressures all meet the temperature axis at \(-273\\ ^\\circ\\text{C}\).
- At constant pressure the density of a gas decreases as its temperature rises, which is why hot air is used in balloons for meteorological purposes.
- Absolute zero, \(-273\\ ^\\circ\\text{C}\), is the temperature at which molecular motion is taken to cease. The gas equation predicts zero volume there, but this cannot happen, since matter must have some definite mass and volume; all gases liquefy or solidify first.
- The Kelvin or absolute scale has its zero at \(-273.15\\ ^\\circ\\text{C}\) (taken as \(-273\\ ^\\circ\\text{C}\) in calculations) and degrees the same size as Celsius degrees. No degree sign is written with it, and all its values are positive.
- \(\\text{K} = ^\\circ\\text{C} + 273\), and \(^\\circ\\text{C} = \\text{K} - 273\).
- Combining Boyle’s law and Charles’ law gives the gas equation, used when both pressure and temperature change. \[\\frac{P\_1V\_1}{T\_1} = \\frac{P\_2V\_2}{T\_2}\]
- Standard temperature and pressure (S.T.P.) are \(0\\ ^\\circ\\text{C}\) or \(273\\ \\text{K}\), and \(760\\ \\text{mm Hg} = 76\\ \\text{cm Hg} = 1\\ \\text{atm}\). The volume of a gas must always be stated along with its temperature and pressure.
- A gas collected over water is moist. Its total pressure is the sum of the partial pressure of the dry gas and the aqueous tension, so \[\\text{actual pressure of gas} = \\text{total pressure} - \\text{aqueous tension}\]
7.14 Key Terms
- Absolute zero — the temperature, \(-273\\ ^\\circ\\text{C}\) or \(0\\ \\text{K}\), at which molecular motion is taken to cease completely; no lower temperature is possible.
- Aqueous tension — the partial pressure exerted by the water vapour present in a gas collected over water.
- Atmospheric pressure — the pressure exerted by the air of the atmosphere on the surface of the earth.
- Boyle’s law equation — the relation \(P\_1V\_1 = P\_2V\_2\), holding for a fixed mass of a dry gas at constant temperature.
- Celsius scale — the everyday temperature scale, once called the centigrade scale, whose zero is placed at the melting point of ice.
- Charles’ law equation — the relation \(\\dfrac{V\_1}{T\_1} = \\dfrac{V\_2}{T\_2}\), holding for a fixed mass of a dry gas at constant pressure.
- Diffusion — the gradual mixing of two substances kept in contact, brought about by the motion of their molecules.
- Gas equation — the relation \(\\dfrac{P\_1V\_1}{T\_1} = \\dfrac{P\_2V\_2}{T\_2}\), used when the pressure and the temperature of a fixed mass of gas both change.
- Gaseous state — the state of matter in which inter-particle attraction is weak and inter-particle space so large that the particles move freely through all the space available.
- Isobar — a graph drawn for a gas at constant pressure, such as the volume–temperature graph of Charles’ law.
- Isotherm — a graph drawn for a gas at constant temperature, such as the volume–pressure graph of Boyle’s law.
- Kinetic energy — the energy a body possesses by virtue of its motion.
- Kinetic molecular theory — the explanation of the behaviour and properties of gases in terms of the constant random motion of their molecules.
- Liquefaction — the change of a gas into a liquid, brought about by cooling it and applying pressure to it.
- Partial pressure — the pressure a gas in a mixture would exert if it alone filled the containing vessel at the same temperature.
- Standard temperature and pressure (S.T.P.) — the agreed reference conditions of \(0\\ ^\\circ\\text{C}\) (\(273\\ \\text{K}\)) and \(760\\ \\text{mm Hg}\) (1 atm), to which gas volumes are referred.
- Standard variables — the three quantities used to describe the physical behaviour of a gas: volume, pressure and temperature.
7.15 Points To Ponder
- Boyle’s law and Charles’ law were both discovered by measuring gases, yet neither law mentions which gas was being measured. What does it tell you about matter that hydrogen, oxygen and carbon dioxide should all obey exactly the same relation?
- Charles’ law, taken literally, predicts that a gas would shrink to nothing at \(-273\\ ^\\circ\\text{C}\). A prediction that cannot come true is usually a sign that a law has been pushed outside the conditions it was built for. Which condition is being broken here, and what happens to the gas long before that temperature is reached?
- The kinetic theory treats gas molecules as taking up almost no room and attracting one another almost not at all. Under what conditions of pressure and temperature would you expect those two assumptions to start failing, and what would happen to the gas then?
- Solids and liquids also expand when heated, but no simple law like Charles’ law is used for them at this level. What is it about the arrangement of particles in a gas that makes its behaviour so much easier to state as a rule?
- A gas collected over water must have its aqueous tension subtracted before the gas laws are applied. Suppose a student forgets to do this. Will the volume they calculate at S.T.P. be too large or too small, and why?
- The Celsius scale places its zero where ice melts, a point that has nothing to do with gases. Would the gas laws have been discovered sooner, later, or in a different form if the Kelvin scale had come first?
7.16 Objective Questions
- The graph of \(V\) against \(\\frac{1}{P}\) for a fixed mass of a dry gas at constant temperature is: [1]
- a hyperbola
- a straight line passing through the origin
- a straight line parallel to the x-axis
- a parabola
- The absolute temperature corresponding to \(-23\\ ^\\circ\\text{C}\) is: [1]
- 223 K
- 250 K
- 273 K
- 296 K
- At constant temperature, if the volume of a fixed mass of a gas is reduced to one-fourth, its pressure becomes: [1]
- one-fourth
- one-half
- two times
- four times
- Standard pressure is: [1]
- 76 mm Hg
- 760 cm Hg
- 76 cm Hg
- 7.6 atm
- A graph plotted for a gas at constant pressure is called: [1]
- an isotherm
- an isobar
- an isotope
- a hyperbola
- Aqueous tension is the pressure exerted by: [1]
- the dry gas alone
- the atmosphere
- the water vapour present in a moist gas
- the mercury column of a barometer
- For a fixed mass of a dry gas at constant temperature, the quantity that stays constant is: [1]
- \(V\)
- \(P\)
- \(PV\)
- \(\\dfrac{V}{T}\)
- For questions a to c, choose the answer from the codes given below.
- Both A and R are true and R is the correct explanation of A.
- Both A and R are true but R is not the correct explanation of A.
- A is true but R is false.
- A is false but R is true.
Assertion (A): All values on the Kelvin scale are positive. Reason (R): The zero of the Kelvin scale is placed at the lowest temperature that can exist. [1]
Assertion (A): A gas collected over water exerts a greater pressure than the same dry gas would. Reason (R): The water vapour in the jar exerts a partial pressure of its own. [1]
Assertion (A): A sealed rigid cylinder of gas left in strong sunlight shows a rise in pressure. Reason (R): The volume of the gas increases on heating. [1]
- Match the following. [5]
| Column A | Column B |
|---|---|
| a. torr | i. graph of Boyle’s law |
| b. isotherm | ii. unit of pressure |
| c. 273 K | iii. fraction appearing in Charles’ law |
| d. \(\\frac{1}{273}\) | iv. ice point |
| e. \(\\text{dm}^3\) | v. unit of volume |
Fill in the blanks. [4]
- At constant pressure, the volume of a fixed mass of a dry gas is ______ proportional to its absolute temperature.
- The partial pressure of the water vapour present in a moist gas is called ______.
- The SI unit of pressure is the ______.
- Theoretically, the volume of a gas would become zero at ______ \(^\\circ\\text{C}\).
State whether each statement is true or false. [4]
- A gas exerts a greater pressure on the base of a vessel than on its sides.
- One millilitre and one cubic centimetre are equal volumes.
- One kelvin is the same size as one degree Celsius.
- Boyle’s law holds for a gas to which more gas is being added.
Name the following. [3]
- The gradual mixing of two gases kept in contact, brought about by molecular motion.
- The temperature scale whose zero is placed at absolute zero.
- The term used for a graph plotted at a constant temperature.
Give one word or phrase for each of the following. [3]
- The pressure exerted by the air of the atmosphere on the surface of the earth.
- The energy a body possesses by virtue of its motion.
- The agreed conditions of temperature and pressure to which gas volumes are referred.
Odd one out, and give a reason: atm, torr, cm Hg, \(\\text{dm}^3\). [2]
Odd one out, and give a reason: 760 mm Hg, 76 cm Hg, 1 atm, 273 K. [2]
7.17 Check Your Concepts
Define diffusion, and give one example of it. [2]
State Boyle’s law, and write its mathematical form. [2]
Why is no degree sign written along with a temperature on the Kelvin scale? [2]
State the three standard variables used to describe the behaviour of a gas, and give the SI unit of each. [2]
Give a reason: gases have a much lower density than solids or liquids. [3]
Give a reason: mountaineers have to carry oxygen cylinders with them. [3]
Differentiate between an isotherm and an isobar, stating the basis of difference. [3]
Differentiate between a dry gas and a moist gas, stating the basis of difference. [3]
Explain Boyle’s law on the basis of the kinetic theory of gases. [4]
Why does a gas fill completely the vessel in which it is kept, while a liquid does not? [4]
What would happen to the volume of a fixed mass of a gas, at constant pressure, as it is cooled towards \(-273\\ ^\\circ\\text{C}\)? Explain why this prediction can never be tested in practice. [4]
- State Charles’ law. (b) Starting from \(V = V\_0\\left(\\dfrac{273 + t}{273}\\right)\), show that \(\\dfrac{V\_1}{T\_1} = \\dfrac{V\_2}{T\_2}\) at constant pressure. [5]
Explain why the temperature and the pressure must always be stated along with the volume of a gas, and how the use of S.T.P. removes the difficulty. [5]
7.18 Equations, Conversions And Identification
Identify the labelled parts of the apparatus shown in Fig. 7.5-1, and state the purpose of the weights, the taped nozzle and the rubber stopper. [3]
Name the method of collection shown in Fig. 7.12-1, and state one property a gas must have if it is to be collected in this way. [2]
State what you would observe when the weights resting on the piston in Fig. 7.5-1 are removed one by one, and give the reason for your observation. [3]
Name the graph shown in Fig. 7.6-1, and state what its extension backwards to the temperature axis shows. [2]
How would you obtain each of the following? Show the working. [3]
- A pressure in cm Hg, from a pressure of 0.5 atm.
- A temperature in kelvin, from a temperature of \(27\\ ^\\circ\\text{C}\).
- A volume in \(\\text{cm}^3\), from a volume of \(0.35\\ \\text{dm}^3\).
Identify the law being verified in each case. [3]
- A graph of \(PV\) against \(P\) gives a straight line parallel to the pressure axis.
- A graph of volume against temperature in \(^\\circ\\text{C}\) gives a straight line that cuts the temperature axis at \(-273\\ ^\\circ\\text{C}\) when extended.
- A graph of volume against \(\\frac{1}{P}\) gives a straight line through the origin.
Identify the quantity described: it must be subtracted from the measured pressure of a gas collected over water before the gas equation is applied. State the name of the corrected pressure as well. [2]
An inflated balloon is left in bright sunshine and bursts. State what you would observe before it bursts, and give the reason. [2]
A cylinder of gas is fitted with a freely moving piston and is then heated. State what you would observe, and state what you would have observed instead if the piston had been locked in place. [3]
7.19 Practice With Numericals
A gas occupies \(300\\ \\text{cm}^3\) at \(750\\ \\text{mm Hg}\). Find its volume at \(500\\ \\text{mm Hg}\), the temperature remaining constant. [3]
\(500\\ \\text{cm}^3\) of a gas at \(27\\ ^\\circ\\text{C}\) is heated to \(127\\ ^\\circ\\text{C}\) at constant pressure. Calculate its new volume. [3]
A gas occupies \(1.5\\ \\text{L}\) at \(2\\ \\text{atm}\) and \(300\\ \\text{K}\). Calculate the volume it will occupy at \(1\\ \\text{atm}\) and \(400\\ \\text{K}\). [3]
At constant temperature, the pressure of a gas is \(900\\ \\text{mm Hg}\). Calculate its new pressure if the volume is decreased by 25 per cent of its original value. [3]
To what temperature, in \(^\\circ\\text{C}\), must \(400\\ \\text{cm}^3\) of a gas at \(27\\ ^\\circ\\text{C}\) be cooled at constant pressure so that it occupies \(300\\ \\text{cm}^3\)? [3]
A gas at S.T.P. occupies \(200\\ \\text{cm}^3\). At what temperature will it occupy \(250\\ \\text{cm}^3\) at the same pressure? Give your answer in kelvin and in \(^\\circ\\text{C}\). [3]
Two vessels of capacity \(3\\ \\text{dm}^3\) and \(2\\ \\text{dm}^3\) contain the same gas at \(400\\ \\text{mm Hg}\) and \(900\\ \\text{mm Hg}\) respectively. They are joined by a tube fitted with a tap. Calculate the final pressure when the tap is opened, the temperature remaining constant. [4]
A cylinder of fixed volume contains a gas at \(27\\ ^\\circ\\text{C}\) and \(2\\ \\text{atm}\). Calculate the pressure of the gas when it is heated to \(87\\ ^\\circ\\text{C}\). [3]
\(60\\ \\text{cm}^3\) of a gas is collected over water at \(15\\ ^\\circ\\text{C}\) and \(748\\ \\text{mm Hg}\). The aqueous tension at \(15\\ ^\\circ\\text{C}\) is \(13\\ \\text{mm Hg}\). Calculate the volume of the dry gas at S.T.P. [4]
Convert: (i) \(57\\ ^\\circ\\text{C}\) to kelvin, (ii) \(350\\ \\text{K}\) to \(^\\circ\\text{C}\), (iii) \(1.5\\ \\text{atm}\) to mm Hg. [3]