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- Question : 1P - (a) Show that, for two large systems in thermal contact, the number .0/.E.0/,E1/ of Section 1.2 can be expressed as a Gaussian in the variable E1. Determine the root-mean-square deviation of E1 from the mean value E1 in terms of other quantities pertaining to the problem. (b) Make an explicit evaluation of the root-mean-square deviation of E1 in the special case when the systems A1 and A2 are ideal classical gases.
- Question : 2P - Assuming that the entropy S and the statistical number of a physical system are related through an arbitrary functional form S D f ./, show that the additive character of S and the multiplicative character of necessarily require that the function f ./ be of the form(1.2.6).
- Question : 3P - Two systems A and B, of identical composition, are brought together and allowed to exchange both energy and particles, keeping volumes VA and VB constant. Show that the minimum value of the quantity .dEA=dNA/ is given by ATB ?? BTA TB ??TA , where the
- Question : 4P - In a classical gas of hard spheres (of diameter D), the spatial distribution of the particles is no longer uncorrelated. Roughly speaking, the presence of n particles in the system leaves only a volume .V ??nv0/ available for the .nC1/th particle; clearly, v0 would be proportional to D3. Assuming that Nv0 V, determine the dependence of .N,V,E/ on V (compare to equation (1.4.1)) and show that, as a result of this, V in the ideal-gas law (1.4.3) gets replaced by .V ??b/, where b is four times the actual volume occupied by the particles.
- Question : 5P - Read Appendix A and establish formulae (1.4.15) and (1.4.16). Estimate the importance of the linear termin these formulae, relative to the main term. =6/" 3=2, for an oxygen molecule confined to a cube of side 10 cm; take " D 0.05 eV.
- Question : 6P - A cylindrical vessel 1mlong and 0.1min diameter is filled with a monatomic gas at P D 1 atm and T D 300 K. The gas is heated by an electrical discharge, along the axis of the vessel, which releases an energy of 104 joules. What will the temperature of the gas be immediately after the discharge?
- Question : 7P - Study the statistical mechanics of an extreme relativisitic gas characterized by the single-particle energy states ".nx,ny,nz/ D hc 2L n2x Cn2y Cn2z 1=2 , instead of (1.4.5), along the lines followed in Section 1.4. Show that the ratio CP=CV in this case is 4/3, instead of 5/3.
- Question : 8P - Consider a system of quasiparticles whose energy eigenvalues are given by ".n/ D nh ; n D 0, 1, 2, : : : .Obtain an asymptotic expression for the number of this system for a given number N of the quasiparticles and a given total energy E. Determine the temperature T of the system as a function of E=N and h , and examine the situation for which E=.Nh / 1.
- Question : 9P - Making use of the fact that the entropy S.N,V,E/ of a thermodynamic system is an extensive quantity, show that N @S @N V,E CV @S @V N,E CE @S @E N,V D S. Note that this result implies that .??N CPV CE/=T D S, that is, N D E CPV ??TS.
- Question : 10P - A mole of argon and a mole of helium are contained in vessels of equal volume. If argon is at 300 K, what should the temperature of helium be so that the two have the same entropy?
- Question : 11P - Four moles of nitrogen and one mole of oxygen at P D 1 atm and T D 300K are mixed together to formair at the same pressure and temperature. Calculate the entropy of mixing per mole of the air formed.
- Question : 12P - Show that the various expressions for the entropy of mixing, derived in Section 1.5, satisfy the following relations: (a) For all N1,V1,N2, and V2, .1S/1 2 D f.1S/??.1S/ g 0, the equality holding when and only when N1=V1 D N2=V2. (b) For a given value of .N1 CN2/, .1S/ .N1 CN2/kln2, the equality holding when and only when N1 D N2.
- Question : 13P - If the two gases considered in the mixing process of Section 1.5 were initially at different temperatures, say T1 and T2, what would the entropy of mixing be in that case?Would the contribution arising from this cause depend on whether the two gases were different or identical?
- Question : 14P - Show that for an ideal gas composed of monatomic molecules the entropy change, between any two temperatures, when the pressure is kept constant is 5/3 times the corresponding entropy change when the volume is kept constant. Verify this result numerically by calculating the actual values of .1S/P and .1S/V per mole of an ideal gas whose temperature is raised from 300 K to 400 K.
- Question : 15P - We have seen that the (P, V)-relationship during a reversible adiabatic process in an ideal gas is governed by the exponent , such that PV D const. Consider a mixture of two ideal gases, with mole fractions f1 and f2 and respective exponents 1 and 2. Show that the effective exponent for the mixture is given by 1 ??1 D f1 1 ??1 C f2 2 ??1
- Question : 16P - Establish thermodynamically the formulae V @P @T D S and V @P @ T D N. Express the pressure P of an ideal classical gas in terms of the variables and T, and verify the above formulae.

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