What went wrong when the king rode under the arch?

It was too low and knocked off his crown

How does the chain of blame unfold in the poem?

Each accused person blames another, ending with the king's own changes

State the basic assumptions of kinetic theory.

A gas consists of a large number of molecules in random motion; intermolecular forces are negligible except during collisions, which are elastic and of negligible duration.

Write the kinetic expression for the pressure of an ideal gas.

\(P=\tfrac13 nm\overline{v^{2}}\), where \(n\) is the number density, \(m\) the molecular mass and \(\overline{v^{2}}\) the mean square speed.

State the kinetic interpretation of temperature.

The average translational kinetic energy of a molecule is \(\tfrac32 k_B T\), proportional to the absolute temperature and independent of the nature of the gas.

Write the formula for the rms speed of gas molecules.

\(v_{rms}=\sqrt{\dfrac{3k_B T}{m}}=\sqrt{\dfrac{3RT}{M}}\); lighter molecules move faster at the same temperature.

How many degrees of freedom has a monatomic and a diatomic gas molecule?

A monatomic molecule has 3 (translational); a rigid diatomic molecule has 5 (3 translational and 2 rotational).

Give the molar specific heats of a monatomic ideal gas.

\(C_v=\tfrac32 R\), \(C_p=\tfrac52 R\) and \(\gamma=C_p/C_v=5/3\).

Define mean free path and give its formula.

The average distance travelled between successive collisions, \(l=\dfrac{1}{\sqrt2\,\pi n d^{2}}\), where \(n\) is number density and \(d\) the molecular diameter.

State Avogadro's hypothesis and how kinetic theory justifies it.

Equal volumes of all gases at the same temperature and pressure contain equal numbers of molecules; kinetic theory shows the number density is the same for all gases under these conditions.

Why do lighter gases diffuse faster?

At the same temperature lighter molecules have a higher rms speed, so their rate of diffusion is greater (inversely proportional to the square root of molecular mass).

Kinetic Theory

Q: State the law of equipartition of energy.

In thermal equilibrium the total energy is shared equally among all degrees of freedom, each contributing \(\tfrac12 k_B T\) (a vibrational mode contributes \(k_B T\)).

Class 11 Physics · Physics Part II · Chapter 5

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CBSE Class 11 · Physics

Ch 8–14 formulas and table values, by page

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Summary

Kinetic theory explains the behaviour of gases by treating them as large numbers of tiny molecules in rapid, random motion, with intermolecular forces negligible except during brief elastic collisions. Matter is made of atoms about an angstrom in size; in gases the molecules are far apart, so the mean free path is large. Gases at low pressure and high temperature obey the ideal-gas equation \(PV=\mu RT=k_B NT\), which embodies Boyle's law, Charles' law, Avogadro's hypothesis and Dalton's law of partial pressures. From the assumption of elastic molecular collisions with the walls, the pressure of an ideal gas is derived as \(P=\tfrac13 nm\overline{v^{2}}\), leading to the kinetic interpretation of temperature: the average translational kinetic energy of a molecule is \(\tfrac32 k_B T\), independent of the gas, and the root-mean-square speed is \(v_{rms}=\sqrt{3k_B T/m}\). The law of equipartition of energy assigns \(\tfrac12 k_B T\) to each degree of freedom, explaining the molar specific heats of monatomic, diatomic and polyatomic gases and of solids. The mean free path \(l=1/(\sqrt2\,\pi n d^{2})\) is the average distance a molecule travels between collisions and accounts for diffusion, viscosity and conduction in gases.

Molecular nature of matterBehaviour of gases and the ideal-gas equationKinetic theory of an ideal gas and pressureKinetic interpretation of temperature and rms speedLaw of equipartition of energySpecific heat capacity of gases and solidsMean free path

Key terms

Ideal gas

A theoretical gas that obeys \(PV=\mu RT\) exactly at all pressures and temperatures; real gases approach it at low pressure and high temperature.

Boltzmann constant

The constant \(k_B=R/N_A=1.38\times10^{-23}\ \text{J K}^{-1}\) linking molecular energy to temperature.

Root-mean-square speed

The square root of the mean of the squared molecular speeds, \(v_{rms}=\sqrt{3k_B T/m}\).

Degree of freedom

An independent way in which a molecule can possess energy — translational, rotational or vibrational.

Law of equipartition of energy

In equilibrium, energy is shared equally among all degrees of freedom, each contributing \(\tfrac12 k_B T\).

Mean free path

The average distance a molecule travels between successive collisions, \(l=1/(\sqrt2\,\pi n d^{2})\).

Extra questions & answers

Try one from this chapter

In kinetic theory, collisions between gas molecules are assumed to be:

Flashcards & interactive practice