UP204: Practice set-1

1. Find the three equations of state for a system with the fundamental equation

U =

(

v0θ

R2

)

S3

NV

here R, v0 and θ are positive constants. Corroborate that the equations of state arehomogeneous zero order (i.e., that T , P and µ are intensive parameters).

2. For the system of problem 1 find µ as a function of T , V and N .

3. Show by a diagram (drawn on arbitrary scale) the dependence of pressure on volume forfixed temperature for the system of problem 2. Draw two such “isotherms”, correspond-ing to two values of temperature, and indicate which isotherm correspond to the highertemperature.

4. Two particular systems have the following equations of state:

1

T (1)=

3

2RN (1)

U (1)

and1

T (2)=

5

2RN (2)

U (2)

Where R is gas constant. The mole number of the first system is N (1) = 2 and that of thesecond is N (2) = 3. The two systems are separated by a diathermal wall, and the totalenergy of the composite system is 2.5× 103J . What is the internal energy of each systemin equilibrium?

5. Two systems with the equations of state given in problem 4 are separated by a diathermalwall. The respective mole numbers are N (1) = 2 and N (2) = 3. The initial temperaturesare T (1) = 250K and T (2) = 350K. What are the values of U (1) and U (2) after equilibriumhas been established? What is the equilibrium temperature?

6. Two particular systems have the following equations of state:

1

T (1)=

3

2RN (1)

U (1),

P (1)

T (1)= R

N (1)

V (1)

and1

T (2)=

5

2RN (2)

U (2),

P (2)

T (2)= R

N (2)

V (2)

The mole number of first system is N (1) = 0.5 and that of the second is N (2) = 0.75. Thesystems are contained in a closed cylinder, separated by a fixed, adiabatic, and impermeable

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piston. The initial temperatures are T (1) = 200K and T (2) = 300K, and the total volumeis 20 liters. The ”setscrew” which prevents the motion of the piston is then removed, andsimultaneously the adiabatic insulation of the piston is stripped off, so that the pistonbecomes moveable , diathermal, and impermeable. What is energy, volume, pressure, andtemperature of each subsystem when equilibrium is established?

It is sufficient to take R = 8.3J/moleK and to assume the external pressure to be zero.

7. A tank has a volume of 0.1m3 and is filled with He gas at a pressure of 5 × 106Pa. Asecond tank has a volume of 0.15m3 and is filled with He gas at a pressure of 6×106Pa. Avalve connecting the two tanks is opened. Assuming He to be a monatomic ideal gas andthe walls of the tanks to be adiabatic and rigid, find the final pressure of the system.

8. a) If the temperatures within the two tanks of problem 8, before opening the valve, hasbeen T = 300K and 350K, respectively, what would the final temperature be?

b) If the first tank had contained He at an initial temperature of 300K, and the secondhad contained a diatomic ideal gas with c = 5/2 and an initial temperature of 350K, whatwould the final temperature be?

9. A vertical cylinder contains ν moles of a monatomic ideal gas and is closed off by a piston ofmass M and area A. The whole system is thermally insulated. The downward accelerationdue to gravity is g. Initially, the piston is clamped in position so that the gas has a volumeV0 and absolute temperature T0. The piston is now released and after some oscillations,it comes to rest in a final equilibrium position corresponding to a smaller volume V ofthe gas where it has temperature T . Neglect any frictional forces that might prevent thepiston from sliding freely in the cylinder. Neglect also the heat capacities of the piston andthe cylinder. Calculate the final temperature T and volume V in terms of T0, V0, the gasconstant R and the quantities ν, M , A and g.

10. A hole is opened in a wall separating two chemically identical single-component subsystems.Each of the subsystems is also in interaction with a pressure reservoir of pressure P r. Usethe enthalpy minimum principal to show that the conditions of equilibrium are T (1) = T (2)

and µ(1) = µ(2).

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