[SOLVED] Phys 362 Statistical and Thermal Physics: Homework 5

Consider an ideal gas and suppose that a function of interest satisfies
F = P V n
where n is an integer.
a) Determine !F
T

V
and !F
T

P
. Eliminate T to rewrite the results in terms of P
and V .b) Determine values of n for which !F
T

V
=
!F
T

P
.Consider three variables x, y, z that are not independent. This means that they are related
by some function, which could be written z = z(x, y) or y = y(x, z) or x = x(y, z) depending on the choice of independent variables.These functions can all be differentiated with
respect to their variables; the derivatives must be related. This exercise will result in general
relationships between these derivatives that are always satisfied.a) To illustrated this let z = x2y. So z(x, y) = x2y. Find expressions for x = x(y, z) and
y = y(x, z). Determine expressions for
!z
y

x
,
!z
x

y
,
!x
y

z
,
!x
z

y
,
!y
x

z
, and !y
z

x
.According to your results how are !z
y

x
and !y
z

x
related to each other? How
about !z
x

y
and !x
z

y
?b) To prove these relationships for any function, first consider x and y as the independent
variables. Express dz in terms of dx and dy, using appropriate partial derivatives. Note
that this must be done for any function z(x, y), not just the special case above. Then
express dx in terms of dy and dz, using appropriate partial derivatives.Substitute this
into the general expression for dz; this will give an expression for dz in terms of dx and
dz. Use the expression to show that
!z
x

y
!x
z

y
= 1 and (1)
!z
x

y
!x
y

z
=
!z
y

x
. (2)These identities will be important throughout the subject.
c) Check that the identities are valid for the function z(x, y) = x2y.
d) There is nothing special about the order of the variables in Eqs. (1) and (2). You could
permute the variables as x y, y z and z x and the identities must still be valid.
Do this and check the resulting identities for z = x2y.e) Check these identities for an ideal gas, with z P, x T and y V.Consider monoatomic gases. For an ideal gas,
E = 3
2
NkT.
a) Using the energy, E, determine cV and cP for this gas.
b) Explain, in terms of the energy involved in the processes to measure the two heat
capacities, why you would expect cP > cV .