Download PY2104 - Introduction to thermodynamics and Statistical physics

PY2104 - Introduction to thermodynamics and Statistical
physics
Review Problems. Part III: Thermodynamic functions
and equilibrium
1) Consider an ideal gas, whose entropy is given by
U
V
n
σ + 5R ln + 2R ln
S=
2
n
n
where n is the number of moles, R is the universal gas constant, U is the
internal energy, V is the volume and σ is constant. Calculae the specific
heats at constant pressure and constant volume, cp and cv for this gas.
2) (i) Show that for a perfect gas undergoing adiabatic expansion, pV γ is constant, where γ = cp /cv . (ii) What is the physical reason for the difference
between cp and cv ?
3) Consider a paramagnetic system. From a thermodynamic point of view, the
system is described by its magnetization, M , the magnetic field, B and the
temperature T . ITs equation of state can be written as
M = CB/T ,
where C is constant (known as Curie constant). Its internal energy is
U = −M B .
The increment of work done by the system upon the external environment
is dW = M dB.
(i) Write an expression for the heat input dQ to the system in terms of
thermodynamic variables, M and B (more precisely, dM and dB).
(ii) Find an expression for the differential of the system entropy, dS (as a
function of dM and dB).
(iii) Derive an expression for the entropy, S.
4) Consider a new matter, for which its equation of state is
p=
AT 3
,
V
where p, T and V are the pressure, volume and temperature, respectively,
and A is constant. Furthermore, the internal energy of this matter is
U = BT n ln(V /V0 ) + f (T ) ,
where B, n and V0 are constants, and f (T ) is a function of the temperature
only. What are B and n ?
5) The equation of state of a van-der-waals gas is
a p + 2 (V − b) = RT .
V
(i) The gas undergoes an isothermal expansion from volume V1 to volume
V2 . Calculate the change in Helmholtz free energy.
(ii) Calculate the change in the internal energy.
6) A certain system is found to have a Gibbs free energy given by
ap
G(p, T ) = RT ln
(RT )5/2
where a and R are constants. Find the specific heat at constant pressure, cp .
7) Consider a thermally isolated system, consisting of two parts, A and B separated by a thermally conducting and movable partition. The volume of part
A is V and that of part B is 2V . The system is filled with ideal gas: in
part A the gas has pressure p and temperature T , while in part B it has
temperature T but pressure 3p. The partition is allowed to move (without
the gases in the two sides mixing). When equilibrium is established:
(i) what is the equilibrium temperature ?
(ii) what is the equilibrium pressure ?
(iii) what is the change in the total entropy ?