Peter asked in Science & MathematicsPhysics · 1 decade ago



Suppose you are given the following equation of state between the entropy S, the

volume V, the internal energy U, and the number of particles N of a thermodynamic


S = A[NVU]^(1/3)

where A is a constant. Derive a relation among:

i) U, N, V and T.

ii) the pressure p, N, V and T.

iii) What is the specific heat at constant volume?

Now assume two identical bodies each consisting of a material which obeys the above

equation of state. N and V are the same for both, and they are initially at temperatures

T1 and T2 respectively. They are to be used as a source of work by bringing them to a

common temperature Tf. This process is accomplished by the withdrawal of heat from

the hotter body and the insertion of some fraction of this heat into the colder body,

with the remainder appearing as work.

iv) What is the range of possible final temperatures?

v) What Tf corresponds to the maximum delivered work, and what is this

maximum amount of work?


Einstein’s model of a solid:

i) Calculate the partition function for a harmonic oscillator in 3-dimensions.

i) From this calculate the specific heat for a solid containing identical N atoms?

iii) Describe the low and high temperature behaviour of the specific heat, pointing

out in which temperature regimes you think the model is valid and where it is

not. In the case of where the model is not valid write a brief description (no

more than half a page) as to how you might improve this model.


Consider a system of two particles (number of particles is fixed), each of

which can be in any one of three quantum states, with respective energies 0, ε and 3ε.

The system is at temperature T. What is the partition function if the particles obey

i) Fermi-Dirac statistics?

ii) Bose-Einstein statistics?


2 Answers

  • 1 decade ago
    Favorite Answer

    S = A[NVU]^(1/3) ...... (1)

    where A is a constant. Derive a relation among:

    i) U, N, V and T.

    Use basic relation from exact differential of Maxwell relation.

    T = (dU/dS)v, v outside the bracket indicates derivation of constant volume

    T = (1 / dS/dU)v, S = A[NVU]^(1/3)

    T = 3*U^(2/3) / A[NV]^(1/3)

    3*U^(2/3) = T* A[NV]^(1/3)

    U^(2/3) = T* A[NV]^(1/3) /3

    U^2 = T^3*A^3*NV / 27

    U = [TA]^(3/2)*[NV]^1/2 / 27^(1/2) ...... (2)

    ii) the pressure p, N, V and T.

    -P = (dF/dV)T ..... (3)

    F = Helmholtz free energy

    Let solve for Helmholtz free energy by using its definition.

    F = U - TS ......... (4)


    F = U - TA[NVU]^(1/3) ......... (5)

    (2) --> (5)

    F = [TA]^(3/2)*[NV]^1/2 / 27^(1/2) - TA[NV]^(1/3) * {[TA]^(3/2)*[NV]^1/2 / 27^(1/2)}^1/3 .. (6)

    (6) --> (3)

    -P = (dF/dV)T

    -P = (d{[TA]^(3/2)*[NV]^1/2 / 27^(1/2) - TA[NV]^(1/3)*{[TA]^(3/2)*[NV]^1/2 / 27^(1/2)}^1/3}/dV)T

    -P = [TA]^(3/2)*[N]^1/2 / 2*[27V]^(1/2) - 0.5*TA[N]^(1/3) * {[TA]^(3/2)*[N]^1/2 / 27^(1/2)}^1/3} / V^1/2

    P = 0.5*TA[N]^(1/3) * {[TA]^(3/2)*[N]^1/2 / 27^(1/2)}^1/3} / V^1/2 - [TA]^(3/2)*[N]^1/2 / 2*[27V]^(1/2)

    iii) What is the specific heat at constant volume?

    Cv = (dU/dT)v, by definition

    Cv = (d{[TA]^(3/2)*[NV]^1/2 / 27^(1/2)}/dT)v

    Cv = 3/2*T^(1/2)[A]^(3/2)*[NV]^1/2 / 27^(1/2)

    Cv = {1.5*[A]^(3/2)*[NVT]^1/2} / {27^(1/2)}

    (2) Not familiar with statistical thermodynamic.

  • 4 years ago

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