An Introduction to Thermal Physics
1st Edition
ISBN: 9780201380279
Author: Daniel V. Schroeder
Publisher: Addison Wesley
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Chapter 8.1, Problem 11P
To determine
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Consider an ideal gas containing N atoms in a container of volume Pressure P, and absolute temperature T1 (not to be confused with K. E. T). Use the virtual theorem to derive the equation of state for a perfect gas.
A. (a) Consider a canonical ensemble having N particle, V volume and at T temperature. Write down the
expression of partition function (Q(N,V,T)) of this canonical ensemble in terms of the microstate energy
Ej.
(b) Write down the expression for Helmhotz free energy (A) and pressure (P) in terms of Q(N,V.T).
(c) Now, assume that for a system of dense gas you can write down the Q(N,V,T) as,
1 (2amk,T
Q(N,V,T)=
N!
(V- Nb)" e
Treat a and b as constants. Get the expression for pressure (P) in terms of V, a, b, N, kg and T. Rearrange
that expression to get a form where in the RHS of the equation will have Nk T. Identify the equation.
calculate the heat capacity of an Einstein solid in the low-temperature limit. Sketch the predicted heat capacity as a function of temperature. (Note: Measurements of heat capacities of actual solids at low temperatures do not confirm the prediction that you will make in this problem.
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- Consider N identical harmonic oscillators (as in the Einstein floor). Permissible Energies of each oscillator (E = n h f (n = 0, 1, 2 ...)) 0, hf, 2hf and so on. A) Calculating the selection function of a single harmonic oscillator. What is the division of N oscillators? B) Obtain the average energy of N oscillators at temperature T from the partition function. C) Calculate this capacity and T-> 0 and At T-> infinity limits, what will the heat capacity be? Are these results consistent with the experiment? Why? What is the correct theory about this? D) Find the Helmholtz free energy from this system. E) Derive the expression that gives the entropy of this system for the temperature.arrow_forwardFind the number density N/V for Bose-Einstein condensation to occur in helium at room temperature (293 K). Compare your answer with the number density for an ideal gas at room temperature at 1 atmosphere pressure.arrow_forwardCompute the quantum volume for an N2 molecule at room temperature, and argue that a gas of such molecules at atmospheric pressure can be treated using Boltzmann statistics. At about what temperature would quantum statistics become relevant for this system (keeping the density constant and pretending that the gas does not liquefy)?arrow_forward
- Could you sketch a distribution function of gaseous with respect to liquid water as a function of temperature (no formula, just a sketch)arrow_forwardSuppose an Einstein Solid is in equilibrium with a reservoir at some temperature T. Assume the ground state energy is 0, the solid is composed of N oscillators, and the size of an energy "unit" is e. (a) Find the partition function for a single oscillator in the solid, Z1. Hint: use the general series summation formula 1+ x + x? + x³ + ... = 1/ (1- x) (b) Find an expression for , the average energy per oscillator in the solid, in terms of kT and e. (c) Find the total energy of the solid as a function of T, using the expression from part (b). (d) Suppose e = 2 eV and T = 25°C. What fraction of the oscillators is in the first excited state, compared to the ground state (assuming no degeneracies of energy levels)?arrow_forwardWhat are the two major assumptions that are made in deriving the partition function for the ideal gas? Do you expect these assumptions to work better for a dilute or dense gas? Explain.arrow_forward
- Show that an irreversible isothermal expansion of an ideal gas against constant external pressure (P2), from an initial state of P1, V1, T to P2, V2, T , with P2 less than P1, is a spontaneous process by showing that ASuniv is positive. You will need to calculate AS SUS and AS surr - upload your calculations for each separately below. You may show that ASuniv is positive graphically. (By calculate, I mean to show that the signs and relative magnitudes of AS and ASsurr are appropriate. A sketch of the graph is ok.) sysarrow_forward5. Non-harmonic gas: let us re-examine the generalized ideal gas introduced in the previous section, using statistical mechanics rather than kinetic theory. Consider a gas of N non-interacting atoms in a d-dimensional box of "volume" V, with a kinetic energy H =EAPI". where i, is the momentum of the ith particle.arrow_forward(a) The mean free path for a classical gas is 1 l = Nad' give a heuristic derivation of the mean free path explaining all the terms and any assump- tions made. (b) The Boltzmann transport equation for a classical distribution function is +ở .V7ƒ +ã · Võƒ = collisions Briefly sketch how it is derived and explain the terms. (c) Explain the significance of the relaxation time Te, and use the relaxation time approxi- mation to rewrite the Boltzmann transport equatio.arrow_forward
- . An ideal classical gas composed of N particles, each of mass m, is enclosed in a vertical cylinder of height L placed in a uniform gravitational field (of acceleration g) and is in thermal equilibrium; ultimately, both N and L → ∞. Evaluate the partition function of the gas and derive expressions for its major thermodynamic properties. Explain why the specific heat of this system is larger than that of a corresponding system in free space.arrow_forwardConsider a van der Waal's gas that undergoes an isothermal expansion from volume V₁ to volume V₂. Calculate the change in the Helmholtz free energy. 2.2 (a) (b) From the theory of thermodynamics, with T and V T independent, ()₁ = T()-p. Show that the change in internal energy is AU = a(-1/2).arrow_forwardConsider the normal modes of a linear chain in which the force constants between nearest- neighbor atoms are alternately C and 10C. Let the masses be equal, and let the nearest-neighbor separation be a/2. Find w(K) at K = 0 and K = TT aarrow_forward
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