Themes › Theme B The particulate nature of matter
B.3Gas laws
Squeeze a balloon and it pushes back harder; leave a ball in the cold and it goes soft. Gases follow simple, exact laws that link their pressure, volume and temperature. This topic finds those laws by experiment, combines them into one equation, and then explains them with a model of tiny molecules bouncing around. There is no additional HL content in B.3, but B.4 (HL) builds directly on it.
Knowledge and science
Nature of science
Experiments and patterns. Robert Boyle (1662) found that squeezing trapped air into half the space doubled its pressure. Jacques Charles and Joseph Gay-Lussac later found how volume and pressure change with temperature. These "empirical" laws came from measurements, before anyone could explain them.
Models. In the 1800s, Maxwell and Boltzmann showed that the same laws follow from a model of molecules as tiny, hard, randomly moving particles obeying Newton's laws. The empirical route and the theoretical route arrive at the same equation, which gives us more confidence in both.
Shared endeavour. Many scientists in different countries contributed: Avogadro in Italy, Clapeyron and Gay-Lussac in France, Boltzmann in Austria and Maxwell in Scotland. Some of them stated the same laws in different but equivalent ways.
Ideal and real. No real gas is perfectly ideal. Physicists use the ideal model because it is simple and works well under the right conditions, and they know when it breaks down.
ToK: questions to think about
- When is an idealised model "good enough" to count as knowledge? The ideal gas law is never exactly true, yet engineers rely on it every day. Is knowledge about an imaginary gas real knowledge?
- Can we know that something exists before we can see it? Around 1900, some respected scientists still doubted that atoms were real, even though the kinetic theory worked so well. What finally counts as evidence for something invisible?
- Do two routes to the same answer make it more certain? The gas laws were found by experiment and also derived from a theory. Does agreement between reason and observation prove anything?
- Is a "law" in science the same as a law in society? Boyle's "law" is broken by every real gas at high pressure. What do scientists mean when they call something a law?
1. Pressure
$P$ is the pressure in pascals (Pa), $F$ is the force acting perpendicular (at right angles) to a surface in N, and $A$ is the area in $\text{m}^2$. $1\ \text{Pa} = 1\ \text{N m}^{-2}$.
Pressure is a scalar: a gas pushes equally in all directions on every surface it touches. Normal air pressure at sea level is about $1.0 \times 10^5$ Pa (100 kPa). That's the weight of about 10 tonnes on every square metre, but we don't notice it because the air inside our bodies and homes pushes back just as hard.
Worked example: why a can collapses
A sealed metal box has one face of area $0.050\ \text{m}^2$. Air is pumped out until the pressure inside is $2.0 \times 10^4$ Pa. The air outside is at $1.0 \times 10^5$ Pa. Find the resultant force on the face.
Pressure difference: $1.0 \times 10^5 - 2.0 \times 10^4 = 8.0 \times 10^4$ Pa.
$F = PA = 8.0 \times 10^4 \times 0.050 = 4000$ N, pushing inwards. That's about the weight of a 400 kg object, which is why thin cans crush when the air inside is removed.
2. The mole and the Avogadro constant
Gases contain enormous numbers of molecules, so we count them in moles. One mole is $6.02 \times 10^{23}$ particles: the Avogadro constant $N_A$.
$n$ is the amount of substance in mol, $N$ is the number of particles, and $N_A = 6.02 \times 10^{23}\ \text{mol}^{-1}$.
The molar mass $M$ is the mass of one mole. In grams, it equals the relative molecular mass from the periodic table: about 4 g for helium, 18 g for water, 28 g for nitrogen (N2) and 44 g for CO2. So the number of moles in a mass $m$ is $n = \dfrac{m}{M}$. Remember to convert grams to kilograms when you need SI units.
Worked example: molecules in a glass of water
How many molecules are there in 0.25 kg of water? (Molar mass of water = 18 g $\text{mol}^{-1}$.)
$n = \dfrac{250\ \text{g}}{18\ \text{g mol}^{-1}} = 13.9$ mol.
$N = nN_A = 13.9 \times 6.02 \times 10^{23} = 8.4 \times 10^{24}$ molecules.
3. The kinetic model of an ideal gas
An ideal gas is a model: an imaginary gas whose molecules obey these assumptions.
- The gas contains a very large number of identical molecules, moving randomly in all directions with a range of speeds.
- The molecules are tiny: their total volume is negligible compared with the volume of the container.
- There are no forces between molecules, except during collisions. So between collisions they move in straight lines at constant speed.
- All collisions, with each other and with the walls, are perfectly elastic: no kinetic energy is lost.
- Collisions last a negligible time compared with the time between them.
- The molecules obey Newton's laws of motion.
With no intermolecular forces, an ideal gas has no intermolecular potential energy: all its internal energy is the kinetic energy of its molecules.
Explaining gas behaviour with molecules
- Pressure: each molecule that hits a wall bounces back, so its momentum changes. By Newton's second and third laws, the wall exerts a force on the molecule and the molecule exerts an equal force on the wall. Billions of collisions every second add up to a steady force on each square metre: the pressure.
- Heating at constant volume: the molecules move faster, so they hit the walls more often and with a bigger change in momentum each time. The pressure rises.
- Compressing at constant temperature: the molecules have less distance to travel between walls, so they hit each square metre more often. The pressure rises, although each collision is just as hard as before.
- Heating at constant pressure: faster molecules would hit harder and more often, so the gas must expand (spreading the collisions over more area and making them less frequent) to keep the pressure the same.
4. The empirical gas laws
For a fixed mass of gas, keep one of $P$, $V$ and $T$ constant and investigate how the other two are related:
- Boyle's law (constant temperature): $P \propto \dfrac{1}{V}$, so $PV = \text{constant}$.
- Charles's law (constant pressure): $V \propto T$, so $\dfrac{V}{T} = \text{constant}$.
- The pressure law (constant volume): $P \propto T$, so $\dfrac{P}{T} = \text{constant}$.
The temperature must be in kelvin. Plotted against °C, the lines in the middle and right graphs don't pass through the origin; extended backwards they cross the axis at −273 °C (absolute zero, see B.1).
In the lab, you take each reading only after waiting for the gas to settle: compressing a gas warms it slightly, and Boyle's law needs the temperature to return to its original value.
5. The ideal gas equation
The three laws combine into one, for a fixed amount of gas:
Experiments show that the constant is proportional to the amount of gas, and is the same for every gas that behaves ideally. This gives the ideal gas equation:
$P$ in Pa, $V$ in $\text{m}^3$, $T$ in K. $n$ is the number of moles and $R = 8.31\ \text{J K}^{-1}\,\text{mol}^{-1}$ is the gas constant. $N$ is the number of molecules and $k_B = 1.38 \times 10^{-23}\ \text{J K}^{-1}$ is the Boltzmann constant. The two forms agree because $N = nN_A$ and $R = N_Ak_B$.
Units trap: volumes are often given in litres or cm³. $1\ \text{litre} = 1000\ \text{cm}^3 = 10^{-3}\ \text{m}^3$.
Worked example: the air in a classroom
A classroom measures 8.0 m × 6.0 m × 3.0 m. The air is at $1.0 \times 10^5$ Pa and 20 °C. Find the amount of air in moles, the number of molecules, and the mass of the air (molar mass of air = 29 g $\text{mol}^{-1}$).
$V = 8.0 \times 6.0 \times 3.0 = 144\ \text{m}^3$ and $T = 293$ K.
$n = \dfrac{PV}{RT} = \dfrac{1.0 \times 10^5 \times 144}{8.31 \times 293} = 5.9 \times 10^3$ mol.
$N = nN_A = 5.9 \times 10^3 \times 6.02 \times 10^{23} = 3.6 \times 10^{27}$ molecules.
Mass $= nM = 5.9 \times 10^3 \times 0.029 = 170$ kg. The air in your classroom weighs about as much as two adults.
Worked example: a weather balloon
A weather balloon holds $5.0\ \text{m}^3$ of helium at $1.0 \times 10^5$ Pa and 290 K at ground level. It rises to a height where the pressure is $2.5 \times 10^4$ Pa and the temperature is 230 K. Find its new volume (assume no helium escapes).
$V_2 = V_1 \times \dfrac{P_1}{P_2} \times \dfrac{T_2}{T_1} = 5.0 \times \dfrac{1.0 \times 10^5}{2.5 \times 10^4} \times \dfrac{230}{290} = 16\ \text{m}^3$.
Quartering the pressure would make it four times bigger, but the cooling reduces this a little. Weather balloons are only partly filled at launch, so they have room to grow.
6. Pressure from molecular collisions
The kinetic model doesn't only explain the gas laws: it predicts the pressure from the motion of the molecules.
Here is the outline of the argument (you need the result and the idea, not the full derivation):
- A molecule of mass $m$ moving at speed $u$ towards a wall bounces back at the same speed. Its momentum changes by $2mu$.
- It travels $2L$ (there and back) before hitting the same wall again, so it hits it every $\dfrac{2L}{u}$ seconds.
- The average force on the wall is the rate of change of momentum: $\dfrac{2mu}{2L/u} = \dfrac{mu^2}{L}$.
- Add up the forces from all $N$ molecules. On average a third of the motion is in each of the three directions, so the total force is $\dfrac{Nm\overline{v^2}}{3L}$, where $\overline{v^2}$ is the mean of the squares of the speeds.
- Divide by the wall's area $L^2$, and note that $\dfrac{Nm}{L^3}$ is the density $\rho$ of the gas.
$\rho$ is the density of the gas in $\text{kg m}^{-3}$ and $\overline{v^2}$ is the mean square speed of its molecules in $\text{m}^2\,\text{s}^{-2}$. The square root of $\overline{v^2}$ is called the root-mean-square (rms) speed, $v_{\text{rms}}$.
Worked example: how fast are air molecules?
Air at $1.0 \times 10^5$ Pa has a density of 1.2 $\text{kg m}^{-3}$. Find the rms speed of its molecules.
$\overline{v^2} = \dfrac{3P}{\rho} = \dfrac{3 \times 1.0 \times 10^5}{1.2} = 2.5 \times 10^5\ \text{m}^2\,\text{s}^{-2}$, so $v_{\text{rms}} = 500\ \text{m s}^{-1}$.
That's faster than the speed of sound (340 $\text{m s}^{-1}$), which makes sense: sound is passed on by the molecules themselves.
7. Temperature, speed and internal energy
Put the two descriptions of a gas together. From $P = \frac{1}{3}\rho\overline{v^2}$ with $\rho = \dfrac{Nm}{V}$, we get $PV = \frac{1}{3}Nm\overline{v^2}$. The ideal gas equation says $PV = Nk_BT$. So:
$$\tfrac{1}{3}m\overline{v^2} = k_BT \qquad\Rightarrow\qquad \tfrac{1}{2}m\overline{v^2} = \tfrac{3}{2}k_BT$$This is the result from B.1: the average kinetic energy of a molecule is $\frac{3}{2}k_BT$. The kinetic model explains why kelvin temperature measures the average kinetic energy of the molecules.
Worked example: helium atoms
Find the rms speed of helium atoms at 300 K. The mass of a helium atom is $6.6 \times 10^{-27}$ kg.
$\overline{v^2} = \dfrac{3k_BT}{m} = \dfrac{3 \times 1.38 \times 10^{-23} \times 300}{6.6 \times 10^{-27}} = 1.9 \times 10^6\ \text{m}^2\,\text{s}^{-2}$, so $v_{\text{rms}} = 1.4 \times 10^3\ \text{m s}^{-1}$.
At the same temperature, all gases have the same average kinetic energy, so lighter molecules move faster. Helium atoms move nearly three times faster than nitrogen molecules.
Internal energy of an ideal gas
An ideal gas has no intermolecular potential energy, so its internal energy $U$ is just the total kinetic energy of its molecules. For a monatomic gas (single atoms, such as helium, neon or argon):
The internal energy of a fixed amount of ideal gas depends only on its temperature, not on its pressure or volume.
Worked example: internal energy of a helium balloon
A party balloon contains 0.50 mol of helium at 20 °C. Find its internal energy.
$U = \tfrac{3}{2}nRT = 1.5 \times 0.50 \times 8.31 \times 293 = 1.8 \times 10^3$ J.
8. Pressure–volume diagrams
The state of a fixed amount of gas is described by its pressure, volume and temperature. Because $PV = nRT$, we only need two of them: a point on a graph of $P$ against $V$ fixes the third. A change of state is drawn as a line or curve on this p–V diagram.
- Constant pressure (isobaric): a horizontal line. Expanding at constant pressure needs the temperature to rise ($V \propto T$).
- Constant volume (isovolumetric or isochoric): a vertical line.
- Constant temperature (isothermal): a curve $PV = \text{constant}$, called an isotherm. Isotherms at higher temperatures lie further from the origin, because $PV = nRT$ is bigger.
In HL B.4 you'll also meet the area under a p–V graph, which is the work done by the gas, and a fourth kind of change, the adiabatic process.
9. Real gases and ideal gases
Real gases break two of the ideal-gas assumptions: their molecules do have some volume, and they do attract each other slightly. This matters when the molecules are close together and moving slowly. That's why real gases can be turned into liquids, while an ideal gas never could.
A real gas behaves almost ideally at:
- low pressure and low density: the molecules are far apart, so their own volume is tiny compared with the container and they rarely feel each other's attraction;
- high temperature (well above the temperature at which it would liquefy): the molecules are so fast that the weak attractions barely affect them.
Air at room temperature and normal pressure is close to ideal. Gases at very high pressure, or cooled close to their boiling point, are not.
10. Common mistakes
- Using °C in any gas equation. Always convert to kelvin. Doubling the Celsius temperature doesn't double the pressure.
- Leaving volumes in litres or cm³. Convert to $\text{m}^3$ before using $PV = nRT$. (In $\dfrac{P_1V_1}{T_1} = \dfrac{P_2V_2}{T_2}$, any units work as long as both sides match, except temperature, which must be in kelvin.)
- Mixing up $n$ and $N$. Use $R$ with moles ($n$) and $k_B$ with numbers of molecules ($N$).
- Saying compressing a gas at constant temperature makes the molecules hit harder. Their speed is unchanged; they hit each part of the wall more often.
- Treating $\overline{v^2}$ as the square of the mean speed. It is the mean of the squared speeds.
- Using $U = \frac{3}{2}nRT$ for real gases or for gases such as O2. It's for ideal monatomic gases only.
- Forgetting that the gas laws need a fixed mass of gas. A leaking tyre doesn't obey $PV/T = \text{constant}$.
11. Check your understanding
The air in a car tyre is at $3.2 \times 10^5$ Pa and 10 °C. After a long drive it reaches 40 °C. Assuming the volume is constant, find the new pressure.
$P_2 = P_1 \times \dfrac{T_2}{T_1} = 3.2 \times 10^5 \times \dfrac{313}{283} = 3.5 \times 10^5$ Pa.
A gas is compressed slowly to a third of its volume at constant temperature. Explain, using the kinetic model, what happens to its pressure.
The pressure triples. The molecules' speeds are unchanged (same temperature), but in a third of the volume they hit each square metre of the walls three times as often.
The kelvin temperature of a gas is doubled. By what factor does the rms speed of its molecules change?
$\overline{v^2} \propto T$, so $\overline{v^2}$ doubles and $v_{\text{rms}}$ increases by a factor of $\sqrt{2} \approx 1.4$.
Two containers hold different ideal gases at the same temperature. Which of these is the same for both: the average kinetic energy of the molecules, or their rms speed?
The average kinetic energy, $\frac{3}{2}k_BT$, depends only on temperature. The rms speed also depends on the mass of the molecules, so lighter molecules are faster.
Under what conditions does a real gas deviate most from ideal behaviour, and why?
At high pressure (high density) and low temperature. The molecules are then close together and slow, so their own volume and the attractive forces between them are no longer negligible.