Kinetic Theory of Gases and Radiation

3.1 Introduction — Equation of State

• The state of a gas is specified by $P$, $V$, $T$, internal energy $E$. The relation between them is the equation of state.

• Boyle: $V \propto 1/P$ at constant $T$. Charles: $V \propto T$ at constant $P$. Gay-Lussac: $P \propto T$ at constant $V$. All for a fixed mass of enclosed gas.

• Combining: $PV \propto T$, i.e. $\dfrac{P_{1}V_{1}}{T_{1}} = \dfrac{P_{2}V_{2}}{T_{2}}$, and $PV = nRT = Nk_{B}T$ with $R = N_{A}k_{B}$.

• The three gas laws hold for real gases only if the pressure is not too high and the temperature is not close to the liquefaction temperature.

• A gas obeying $PV = nRT$ at all pressures and temperatures is an ideal gas.

3.2 Behaviour of a Gas

• $P$, $V$, $T$ are a macroscopic description. There are about $10^{23}$ particles per m$^3$, so describing the gas through the motion of a single molecule (as one would a thrown stone with Newton's laws) is futile.

• Approach of kinetic theory: make assumptions about the molecules, take averages of physical quantities over the enormous number of particles, and relate those averages to the macroscopic parameters.

3.3 Ideal Gas and Real Gas

• A real gas is said to be in the ideal state when its molecules are so far apart that intermolecular interaction is practically absent — that is, at sufficiently low density, meaning low pressure or high temperature, with $T$ well above the liquefaction temperature.

• Deviation is largest at high pressure and low temperature because the molecules are then close together, so both their finite size and their mutual attraction matter.

• Molecules are uniformly dispersed in the container and execute random motion. On approach they repel, behaving as small hard spheres, so collisions are elastic and both speed and direction change abruptly. Between two successive collisions a molecule travels in a straight line with constant velocity; force acts only during a collision.

3.4 Mean Free Path

• Definition: the average distance traversed by a molecule with constant velocity between two successive collisions.

$$\lambda = \dfrac{1}{\sqrt{2}\,\pi d^{2}\left(\dfrac{N}{V}\right)} = \dfrac{k_{B}T}{\sqrt{2}\,\pi d^{2}P}$$

• $\lambda$ varies inversely with the number density $N/V$: higher density, more collisions, smaller $\lambda$.

• $\lambda$ varies inversely with $d^{2}$, not $d$, because collision depends on the cross section of a molecule.

• Application: in metal-film coating, the enclosure is evacuated so that $\lambda$ of the air molecules exceeds the size of the enclosure; the metal vapour atoms then reach the target without colliding with air molecules and get deposited.

• Textbook result: for nitrogen at $0\,^{\circ}$C and 1 atm with $d = 324$ pm, $\lambda = 8.0 \times 10^{-8}$ m, about 247 molecular diameters.

3.5 Pressure of Ideal Gas

$$P = \dfrac{1}{3}\dfrac{N}{V}m\overline{v^{2}} = \dfrac{1}{3}\rho\overline{v^{2}}$$

• Set-up: $n$ moles in a cubical box of side $L$, volume $V = L^{3}$, walls at constant temperature $T$, walls parallel to the coordinate planes. Intermolecular collisions neglected; only elastic collisions with the walls counted.

• Key steps to state: $\Delta p_{x} = -2mv_{x}$, momentum given to the wall $= +2mv_{x}$; distance between two collisions with the same wall $= 2L$, so $\Delta t = 2L/v_{x}$; average force from one molecule $= mv_{x1}^{2}/L$; divide total force by area $L^{2}$; then use $\overline{v_{x}^{2}} = \dfrac{1}{3}\overline{v^{2}}$.

• The result is valid for a container of any shape, and intermolecular collisions do not change it, because in a gas in equilibrium every molecule that loses a particular velocity is replaced by another acquiring it.

3.6 Root Mean Square (rms) Speed

$$v_{rms} = \sqrt{\dfrac{3RT}{M_{0}}} = \sqrt{\dfrac{3k_{B}T}{m}} = \sqrt{\dfrac{3P}{\rho}} \qquad \Rightarrow \qquad v_{rms} \propto \sqrt{T}$$

• Mean square speed $\overline{v^{2}} \propto T$; rms speed $\propto \sqrt{T}$ and $\propto 1/\sqrt{M_{0}}$. At the same temperature, lighter molecules move faster.

• Maxwell's distribution of molecular speeds: a plot of $n_{v}$ against $v$; the shaded strip $n_{v}\,dv$ is the number of molecules with speeds between $v$ and $v + dv$. Speeds range from zero to infinity; averages such as $\overline{v^{2}}$ follow from this distribution.

• Speed of sound $v_{s} = \sqrt{\gamma RT/M_{0}}$ is of the same order as, but always less than, $v_{rms}$ (since $\gamma < 3$). The molecules are the medium that transports sound energy.

3.7 Interpretation of Temperature in Kinetic Theory

$$PV = \dfrac{2}{3}E, \qquad E = \dfrac{3}{2}Nk_{B}T, \qquad \dfrac{E}{N} = \dfrac{3}{2}k_{B}T$$

• The internal energy of an ideal gas is purely kinetic — there is no potential energy term because the molecules are non-interacting.

• Average energy per molecule is proportional to the absolute temperature and is the same for all gases at a given $T$. This is the equation that links the macroscopic parameter $T$ to molecular motion.

• Boyle's law from kinetic theory: $P = \dfrac{2}{3}\dfrac{N}{V}\left(\dfrac{1}{2}m\overline{v^{2}}\right)$. At a fixed temperature $\dfrac{1}{2}m\overline{v^{2}}$ is constant and $N$ is fixed, so $P = \text{constant}/V$, i.e. $P \propto 1/V$.

• Heating a gas: the supplied heat increases the average kinetic energy of the molecules; since $\dfrac{1}{2}m\overline{v^{2}} = \dfrac{3}{2}k_{B}T$, the temperature rises.

3.8 Law of Equipartition of Energy

• Statement: for a gas in thermal equilibrium at temperature $T$, the average energy per molecule associated with each quadratic term (in velocity or in position) is $\dfrac{1}{2}k_{B}T$.

• Degrees of freedom: the total number of coordinates or independent quantities required to describe the position and configuration of the system completely.

• Degrees of freedom depend on the atomicity of the molecule (number of atoms), its geometric structure or shape, and the temperature.

• A diatomic molecule lying along the $x$-axis rotates about the $y$- and $z$-axes only; rotation about the $x$-axis is not a rotation, as it does not change the positions of the atoms.

• Each vibrational mode contributes $2 \times \dfrac{1}{2}k_{B}T = k_{B}T$, because vibrational energy has both a kinetic term $\dfrac{1}{2}mu^{2}$ and a potential term $\dfrac{1}{2}kr^{2}$, each quadratic.

• The law holds at high temperatures and fails at extremely low temperatures where quantum effects become important.

3.9 Specific Heat Capacity

• Two specific heats are defined for a gas because a small rise in temperature causes a considerable change in both volume and pressure; heating at constant volume and at constant pressure are physically different.

• Mayer's relation: $C_{P} - C_{V} = R$ for one mole, or $C_{P} - C_{V} = R/J$ if heat is in calories and work in joules. In terms of principal specific heats, $S_{P} - S_{V} = \dfrac{R}{M_{0}J}$.

• $C_{P} > C_{V}$ always: at constant pressure part of the heat is spent as work $P\,dV$ in moving the piston, so more heat is needed for the same rise in temperature.

3.10 Absorption, Reflection and Transmission of Heat Radiation

• Radiation is the fastest mode of heat transfer and the only one that needs no material medium; physical contact is not required. Thermal radiation is electromagnetic radiation emitted by bodies at a temperature higher than the surroundings.

• $Q = Q_{a} + Q_{r} + Q_{t}$, hence $a + r + t_{r} = 1$. All three coefficients are dimensionless.

• Coefficient of absorption $a$: ratio of the amount of heat absorbed to the total heat incident.

• Coefficient of reflection $r$: ratio of the radiant energy reflected to the total energy incident.

• Coefficient of transmission $t_{r}$: ratio of the radiant energy transmitted to the total energy incident.

• $a$, $r$ and $t_{r}$ depend on the wavelength of the incident radiation as well as on the material: the same body can be diathermanous at one wavelength and a good absorber at another.

3.11 Perfect Blackbody

• Definition: a body which absorbs the entire radiant energy incident on it, so $a = 1$.

• Lamp black and platinum black absorb about 97% of incident radiant heat and so closely resemble a perfect blackbody.

• Ferry's blackbody: a double-walled hollow sphere with a tiny aperture; space between the walls evacuated, outer surface silvered, inner surface coated with lamp black, with a conical projection opposite the aperture. Radiation entering suffers multiple reflections and is almost completely absorbed. The aperture behaves like a perfect blackbody, and its area is the effective area of the blackbody.

• Good absorbers are good emitters; poor absorbers are poor emitters. A poor absorber is a good reflector, so a good reflector is a poor emitter — the reason for silvering the walls of a thermos flask.

• Cavity radiator: a block with an internal cavity connected to the outside by a small hole. Radiation entering cannot escape, so the cavity acts as a blackbody. When heated, it emits cavity radiation whose nature depends only on the temperature of the walls, not on the shape, size or material of the cavity.

3.12 Emission of Heat Radiation

• Prevost's theory of exchange of heat (1792): all bodies at all temperatures above 0 K radiate thermal energy and simultaneously absorb radiation from their surroundings.

• If a body radiates more than it absorbs its temperature falls, and vice versa. When the rate of emission equals the rate of absorption, the temperature is constant and the body is in thermal equilibrium with its surroundings.

• Amount of heat radiated depends on: absolute temperature $T$; the nature of the body (material, polished or not); surface area $A$; and the time $t$ for which it radiates.

• Emissive power (radiant power) $R = \dfrac{Q}{At}$: heat radiated per unit area per unit time. SI unit W m$^{-2}$, dimensions $[L^{0}M^{1}T^{-3}]$.

• Coefficient of emission (emissivity) $e = \dfrac{R}{R_{B}}$, the ratio of the emissive power of the surface to that of a perfect black surface at the same temperature. $e = 1$ for a perfect blackbody, $e = 0$ for a perfect reflector; for copper $e = 0.3$. Emissivity is larger for rough surfaces and smaller for smooth, polished ones, and varies somewhat with temperature and wavelength.

• Colour with temperature: below about $800\,^{\circ}$C the radiation is longer than visible wavelengths (infrared, invisible); around $800\,^{\circ}$C the body appears red; around $3000\,^{\circ}$C it looks white hot, like a tungsten filament.

3.13 Kirchhoff's Law of Heat Radiation

• Statement: at a given temperature, the ratio of the emissive power to the coefficient of absorption of a body is equal to the emissive power of a perfect blackbody at the same temperature, for all wavelengths.

• Equivalent statement: for a body emitting and absorbing thermal radiation in thermal equilibrium, the emissivity is equal to the absorptivity. Symbolically $a = e$, or more specifically $a(\lambda) = e(\lambda)$.

• Consequence: a body with high emissive power also has high absorptive power, and vice versa.

• Everyday applications: the bottom of a cooking utensil is blackened so that it absorbs heat well, while the top is polished so that it emits (and so loses) little; a good reflector such as the silvered wall of a thermos is a poor emitter.

3.14 Spectral Distribution of Blackbody Radiation

• Lummer and Pringsheim measured the radiant power of a cavity radiator at different wavelengths, keeping the source at different fixed temperatures, and plotted $R_{\lambda}$ against $\lambda$. The distribution depends only on the absolute temperature and not on the material.

• Diagram in words: a family of humped curves, $R_{\lambda}$ on the $y$-axis against $\lambda$ on the $x$-axis, one curve per temperature. Each rises from zero, peaks at $\lambda_{max}$, then falls asymptotically. Higher-temperature curves lie entirely above lower-temperature ones and their peaks are shifted to shorter wavelengths (to the left).

• Observations from the curves: (i) the energy is not uniformly distributed over the spectrum; (ii) with increasing $\lambda$ the radiant power first increases, reaches a maximum and then decreases; (iii) the area under a curve gives the total energy emitted per unit time per unit area at all wavelengths; (iv) $\lambda_{max}$ decreases as $T$ increases; (v) the total radiant power increases with temperature; (vi) at 300 K the most intense wave has $\lambda \approx 5 \times 10^{-5}$ m, in the infrared.

• Wien's displacement law: the wavelength for which the emissive power of a blackbody is maximum is inversely proportional to the absolute temperature: $\lambda_{max}T = b$, with $b = 2.897 \times 10^{-3}$ m K. Used to find the temperature of distant stars, the Sun and the Moon. White dwarfs have surface temperature about 10000 K, red giants about 3000 K.

• Theories: Wien's expression from thermodynamics fitted only short wavelengths; the Rayleigh-Jeans formula from equipartition fitted long wavelengths but diverged at short wavelengths. Planck combined the two with the hypothesis of quantised oscillator energy $E = nh\nu$, with $h = 6.626 \times 10^{-34}$ J s and $n$ a positive integer, and described the whole spectrum.

3.15 Stefan-Boltzmann Law of Radiation

• Statement: the rate of emission of radiant energy per unit area, or the power radiated per unit area, of a perfect blackbody is directly proportional to the fourth power of its absolute temperature.

$$R = \sigma T^{4}, \qquad \sigma = 5.67 \times 10^{-8}\ \text{W m}^{-2}\text{K}^{-4}$$

• Power radiated by a perfect blackbody depends only on its temperature, not on colour, material or nature of surface. For an ordinary body $R = e\sigma T^{4}$.

• Body at $T$ in surroundings at $T_{0}$: net loss per unit area per unit time $= e\sigma\left(T^{4} - T_{0}^{4}\right)$; if $T < T_{0}$ the body gains $e\sigma\left(T_{0}^{4} - T^{4}\right)$ instead.

• Doubling the absolute temperature increases the radiated power by $2^{4} = 16$ times.

Traps that cost marks

• Always convert $^{\circ}$C to K before using any of $v_{rms}$, K.E., Wien's law or Stefan's law. $127\,^{\circ}$C is 400 K, not 127 K.

• $\lambda_{max}$ is the wavelength of maximum intensity, not the longest wavelength emitted.

• In the mean free path formula the diameter is squared and there is a factor $\sqrt{2}$: $\lambda = \dfrac{1}{\sqrt{2}\pi d^{2}(N/V)}$, not $\dfrac{1}{\sqrt{2}\pi d(N/V)}$.

• Mean speed $\bar{v}$ and rms speed $v_{rms}$ are different quantities; do not square the mean, average the squares.

• For a net rate of loss of heat use $\left(T^{4} - T_{0}^{4}\right)$, not $\left(T - T_{0}\right)^{4}$.

• $\sigma = 5.67 \times 10^{-8}$ carries a negative exponent; a positive exponent is a very common slip.

• Mayer's relation $C_{P} - C_{V} = R$ is per mole. Use $S_{P} - S_{V} = R/(M_{0}J)$ when the data are per unit mass in calories.

• Degrees of freedom for a diatomic molecule are 5 only if it is treated as a rigid rotator; if vibration is included they are 7 quadratic terms.

• $a$, $r$ and $t_{r}$ have no units and no dimensions; do not attach J or W to them.

• In $PV = Nk_{B}T$, $N$ is the number of molecules; in $PV = nRT$, $n$ is the number of moles. Mixing them up changes the answer by $N_{A}$.