doPhysics · Thermal Physics 1
Kinetic Theory of Gases
A gas as a crowd of molecules in fast, random motion. The gas laws and pV = nRT, pressure from molecular hits, temperature as molecular energy, the spread of molecular speeds, degrees of freedom and the heat capacities of gases, the mean free path, real gases and the fall of pressure with height.
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Why this chapter
Why this chapter exists
A tyre hardens in the afternoon sun, a pressure cooker whistles, and dal refuses to cook in an open pot at Leh. The gas laws describe all of this, but they do not explain it. This chapter explains it with one picture: a gas is a crowd of tiny molecules in ceaseless motion, obeying Newton's laws. From that picture come the pressure of a gas and the meaning of temperature. The speeds of the molecules follow, and so does the heat a gas needs to become warmer. The picture also shows its own limits, in real gases and in the thin air of the mountains.
You will learn
By the end of this chapter you will
- Use the laws of Boyle, Charles, Gay-Lussac, Avogadro and Dalton, and the ideal gas equation \(pV = nRT = NkT\).
- Derive the pressure of a gas from molecular hits, \(p = \tfrac13 \rho\overline{v^2}\), and find root mean square speeds from pressure or temperature.
- Explain temperature as the average kinetic energy of the molecules, 3/2 kT, and read the Maxwell distribution with its three characteristic speeds.
- Count the degrees of freedom of a molecule, apply equipartition, and find the internal energy \(U = (f/2)nRT\).
- Find \(C_v\), \(C_p\) and \(\gamma\) for pure gases and mixtures, and explain \(C_p - C_v = R\) by the work of an expanding gas.
- Find the mean free path of a molecule, use the van der Waals equation, and find the pressure at a height with the barometric formula.
Before you begin
Before you begin
From Mechanics you should know momentum, and Newton's second law as a rate of change of momentum. You should know kinetic energy \(\tfrac12 mv^2\) and the work \(Fd\) of a steady force. You should know pressure as force per area, and \(dp = -\rho g\,dh\) from the unit on fluids. From the chapter on temperature you should know the kelvin scale, \(T = t + 273\). Rotational energy \(\tfrac12 I\omega^2\) and escape speed are used once each.
Lessons
1
The Gas Laws and the Ideal Gas Equation
Pressure, volume, temperature and amount describe a gas. Boyle, Charles and Gay-Lussac found how each pair is linked, Avogadro counted the molecules, and the four laws join into one equation, pV = nRT.
Boyle charles and gay lussacAvogadro and the moleThe ideal gas equationDaltons law of partial pressures
2
Pressure from Molecular Collisions
A gas is a crowd of tiny molecules in fast, random motion. Their hits on a wall combine into a steady push, \(p = \tfrac13 \rho\overline{v^2}\), and this one result tells how fast the molecules of air must move.
The ideal gas modelPressure from molecular impactsRoot mean square speed
3
Temperature and Molecular Speeds
Temperature measures the average kinetic energy of the molecules, 3/2 kT each, whatever the gas. The speeds themselves are spread over the Maxwell distribution, with three characteristic speeds, and the fast tail is why light gases leak into space.
Kinetic meaning of temperatureMaxwell speed distributionThree characteristic speedsWhy light gases escape
4
Degrees of Freedom and Equipartition
A molecule can store energy by moving, by tumbling and by vibrating. Each independent way holds \(\tfrac12 kT\) on average, so the internal energy of an ideal gas is \(U = (f/2)nRT\), and it depends on the temperature alone.
Degrees of freedomEquipartition of energyInternal energy of an ideal gas
5
Molar Heat Capacities of Gases
A gas needs more heat to warm at constant pressure than at constant volume, because it also pushes back its surroundings. Counting degrees of freedom gives \(C_v = (f/2)R\), \(C_p = C_v + R\) and \(\gamma = 1 + 2/f\), for pure gases, for mixtures, and \(3R\) for solids.
Molar heat capacity at constant volumeCp cv and gammaMixtures of gasesSolids and dulong petit
6
Mean Free Path, Real Gases and the Atmosphere
Where the simple model meets the real world. A molecule travels only about 0.1 μm between collisions; real molecules have size and attract one another, as van der Waals showed; and the weight of the air makes its pressure fall exponentially with height.
Mean free pathReal gases and van der waalsThe barometric formula
Read in order. The small labels are the topics of each lesson. They take your colour as you answer the checks.
0%The Workbook
Questions to solve on paper, marked by you.
Glossary
The key words of this chapter, with the meaning this chapter gives them.
- absolute zero
- The zero of the kelvin scale, \(-273\,^\circ\text{C}\). The volume of every gas at constant pressure, extended backwards, would reach zero there.
- average speed
- The mean of the molecular speeds, \(\bar v = \sqrt{8RT/\pi M}\). It lies between the most probable speed and the root mean square speed.
- Avogadro constant
- The number of particles in one mole, \(N_A = 6.02 \times 10^{23}\,\text{mol}^{-1}\).
- barometric formula
- The pressure at height \(h\) in an isothermal atmosphere, \(p = p_0\,e^{-Mgh/RT}\).
- Boltzmann constant
- The gas constant per molecule, \(k = R/N_A = 1.38 \times 10^{-23}\,\text{J/K}\). It appears in \(pV = NkT\) and in the energy \(\tfrac12 kT\) of each degree of freedom.
- compressibility factor
- The ratio \(Z = pV/nRT\). It is \(1\) for an ideal gas; below \(1\) attraction dominates, and above \(1\) the size of the molecules dominates.
- critical temperature
- The temperature above which a gas cannot be liquefied by pressure alone; \(304\,\text{K}\) for carbon dioxide.
- degrees of freedom
- The number of independent ways in which a molecule can hold energy. It is \(3\) for a single atom, \(5\) for a diatomic molecule near room temperature and \(6\) for a non-linear molecule.
- equipartition of energy
- The law that in thermal equilibrium each degree of freedom of a molecule holds, on average, an energy \(\tfrac12 kT\).
- escape speed
- The least speed at which a body leaving a planet or moon escapes its gravity for good. It is \(11.2\,\text{km/s}\) for the Earth and \(2.4\,\text{km/s}\) for the Moon.
- ideal gas
- A gas whose molecules are points that exert no forces on one another except in elastic collisions. It obeys \(pV = nRT\) exactly; real gases come close at low pressure and high temperature.
- internal energy
- The total energy of the random motion of the molecules of a system, measured from its centre of mass. For an ideal gas, \(U = \tfrac f2 nRT\), a function of the temperature alone.
- isotherm
- A curve on a graph of pressure against volume along which the temperature is constant. For an ideal gas it is the hyperbola \(pV = \text{constant}\).
- kinetic theory
- The explanation of the properties of a gas, such as pressure and temperature, from the motion of its molecules under Newton's laws.
- law of Dulong and Petit
- At room temperature most solid elements have a molar heat capacity close to \(3R\), about \(25\,\text{J/(mol K)}\).
- Maxwell distribution
- The way the speeds of the molecules of a gas in equilibrium are shared out. Its curve rises from zero to a peak and falls in a long tail, and the area under it is the number of molecules.
- Mayer's relation
- For an ideal gas, \(C_p - C_v = R\). The extra heat at constant pressure pays for the work \(p\,\Delta V = nR\,\Delta T\) done by the expanding gas.
- mean free path
- The average distance a molecule travels between two successive collisions, \(\lambda = 1/(\sqrt2\,\pi d^2 n)\); about \(0.1\,\mu\text{m}\) for air at \(300\,\text{K}\) and \(1\,\text{atm}\).
- molar heat capacity
- The heat needed to raise the temperature of one mole of a substance by one kelvin, \(C = Q/(n\,\Delta T)\), in J/(mol K). For a gas it depends on how the gas is heated.
- molar heat capacity at constant volume
- The molar heat capacity of a gas heated in a fixed volume, \(C_v = \tfrac f2 R\). No work is done, so all the heat raises the internal energy.
- mole
- The amount of substance that contains \(N_A = 6.02 \times 10^{23}\) particles. Its mass in grams is the molar mass.
- most probable speed
- The speed at the peak of the Maxwell distribution, \(v_{\text{mp}} = \sqrt{2RT/M}\): more molecules have speeds near it than near any other.
- number density
- The number of molecules in each unit volume of a gas, \(N/V = p/kT\); about \(2.45 \times 10^{25}\,\text{m}^{-3}\) for air at \(300\,\text{K}\) and \(1\,\text{atm}\).
- partial pressure
- The pressure one gas in a mixture would exert if it filled the whole vessel alone at the same temperature. The partial pressures add to the total pressure.
- ratio of specific heats
- The ratio \(\gamma = C_p/C_v = 1 + 2/f\) of the two molar heat capacities of a gas: \(5/3\) for a monatomic gas, \(7/5\) for a diatomic gas.
- real gas
- An actual gas, whose molecules have size and attract one another. It departs from \(pV = nRT\) at high pressure and low temperature.
- root mean square speed
- The square root of the mean of the squares of the molecular speeds, \(v_{\text{rms}} = \sqrt{\overline{v^2}} = \sqrt{3RT/M}\). A molecule moving at this speed has the average kinetic energy of the gas.
- scale height
- The height \(H = RT/Mg\) over which the pressure of an isothermal atmosphere falls by a factor \(e\); about \(8.8\,\text{km}\) for air at \(300\,\text{K}\).
- van der Waals equation
- The equation of state \((p + an^2/V^2)(V - nb) = nRT\) for a real gas, where \(a\) measures the attraction between molecules and \(b\) their volume per mole.