5.2 A box made of perfectly reflecting walls is divided by a perfectly reflecting partition

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into compartments 1 and 2. Initially a gas at temperature T1 was confined in compartment 1, and compartment 2 was empty. A small hole of dimension much less than the mean free path of the gas is opened in the partition for a short time to allow a small fraction of the gas to escape into compartment 2. The hole is then sealed off and the new gas in compartment 2 comes to equilibrium. (a) During the time when the hole was open, what was the flux dI of molecules crossing into compartment 2 with speed between v and v + dv (b) During the same time, what was the average energy per particle of the molecules crossing into compartment 2 (c) After final equilibrium has been established, what is the temperature Tz in compartment 2 Answer. Tz = ~Tl.

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5.3 (a) Explain why it is meaningless to speak of a sound wave in a gas of strictly

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(b) In view of (a), explain the meaning of a sound wave in an ideal gas.

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5.4 Show that the velocity of sound in a real substauce is to a good approximation given by c = 1/ P"s , where p is the mass density and "s the adiabatic compressibility, by the

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following steps. (a) Show that in a sound wave the density oscillates adiabatically if

K CApC v

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where

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coefficient of thermal conductivity

A = wavelength of sound wave

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p = mass density

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Cv = C

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(b) Show by numerical examples, that the criterion stated in (a) is well satisfied in most

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5.5 A flat disk of unit area is placed in a dilute gas at rest with initial temperature T. Face A of the disk is at temperature T, and face B is at temperature T1 > T (see sketch).

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Molecules striking face A reflect elastically. Molecules striking face B are absorbed by the disk, only to re-emerge from the same face with a Maxwellian distribution of temperature

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Tl

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(a) Assume that the mean free path in the gas is much smaller than the dimension of the

disk. Present an argument to show that after a few collision times the gas can be described by the hydrodynamic equations, with face B replaced by a boundary condition for the temperature. (b) Write down the first-order hydrodynamic equations for (a), neglecting the flow of the gas. Show that there is no net force acting on the disk. (c) Assume that the mean free path is much larger than the dimensions of the disk. Find the net force acting on the disk.

5.6 A square vane, of area 1 cn , painted white on one side, black on the other, is

attached to a vertical axis and can rotate freely about it (see the sketch). Suppose the arrangement is placed in He gas at room temperature and sunlight is allowed to shine on the vane. Explain qualitatively why (a) at high density of the gas the vane does not move; (b) at extremely small densities the vane rotates; (c) at some intermediate density the vane rotates in a sense opposite to that in (b). Estimate this intermediate density and the corresponding pressure.

Gas at temp. T

Fig. P5.5

Fig. P5.6

5.7 A dilute gas, infinite in extension and composed of charged molecules, each of charge

e and mass m, comes to equilibrium in an infinite lattice of fixed ions. In the absence of an external electric field the equilibrium distribution function is

I(O)(p)

n(2'ITmkT)-3/2e-p2/2mkT

where n and T are constants. A weak uniform electric field E is then turned on, leading to a new equilibrium distribution function. Assume that a collision term of the form

a ) =_1-/(0) l ( at coli T

where T is a collision time, adequately takes into account the effect of collisions among molecules and between molecules and lattice. Calculate (a) the new equilibrium distribution function I, to the first order; (b) the electrical conductivity 0, defined by the relation ne(v) = oE

STATISTICAL MECHANICS

CLASSICAL STATISTICAL MECHANICS

6.1 THE POSTULATE OF CLASSICAL STATISTICAL MECHANICS

Statistical mechanics is concerned with the properties of matter in equilibrium in the empirical sense used in thermodynamics. The aim of statistical mechanics is to derive all the equilibrium properties of a macroscopic molecular system from the laws of molecular dynamics. Thus it aims to derive not only the general laws of thermodynamics but also the specific thermodynamic functions of a given system. Statistical mechanics, however, does not describe how a system approaches equilibrium, nor does it determine whether a system can ever be found to be in equilibrium. It merely states what the equilibrium situation is for a given system. We recall that in the kinetic theory of gases the process of the approach to equilibrium is rather complicated, but the equilibrium situation, the MaxwellBoltzmann distribution, is simple. Furthermore, the Maxwell-Boltzmann distribution can be derived in a simple way, independent of the details of molecular interactions. We might suspect that a slight generalization of the method used-the method of the most probable distribution-would enable us to discuss the equilibrium situation of not only a dilute gas but also any macroscopic system. This indeed is true. The generalization is classical statistical mechanics. We consider a classical system composed of a large number N of molecules occupying a large volume V. Typical magnitudes of N and V are molecules V :::: 10 23 molecular volumes Since these are enormous numbers, it is convenient to consider the limiting case N--+oo V --+ 00

N :::: 10 23