2TP - Thermal Physics
Department of Physics, University of Surrey
Problem Set 2:
Thermodynamic Work
You should read Chapters 1 and 2 of Thermal Physics by C.B.P. Finn.
(1) Calculate the work done by one mole of an ideal gas during a quasi-static, isothermal
expansion in which the volume is doubled. How much work is done if the volume is
doubled under constant pressure (i.e. an isobaric process)? Write an expression for the
difference between the work done isothermally and the work done isobarically.
(2) 10 moles of an ideal gas are compressed isothermally and reversibly from a pressure
of 1 atmosphere to 10 atmospheres at 300 K. How much work is done on the gas?
(3) An ideal gas undergoes the following reversible cycle: (a) an isobaric (i.e. at
constant pressure) expansion from the state (V1, P1) to the state (V2, P1); (b) a reduction in
pressure to P2 at constant volume; (c) an isobaric reduction in volume to the state (V1,
P2); and an increase in pressure at constant volume, back to the original state again.
What work is done in this cycle? If P1 = 3 atm., P2 = 1 atm., V1 = 1 L and V2 = 2 L, how
much work is done by the gas in one cycle?
(4) The equation of state for a rubber band is
 L  Lo  2 
  aT      ,
 Lo  L  
where  is the tension on the band, L is its length, Lo is its original length, and a is a
constant equal to 1.3 x 10-2 NK-1. How much work is performed when the band is
stretched isothermally and reversibly from its original length of 10 cm to a final length of
20 cm. The temperature is constant at 20 ºC.
(5)(a) The surface tension of a soap bubble surface is 30 mJ m-2. How much work is
done on the bubble when it is expanded from having a radius of 5 cm to a radius of 15
cm. Assume that the surface tension  does not vary with surface area.
(b) Derive an expression for the work done if  depends on area as:  = o exp(A/k),
where o and k are constants.
(6) Why are soap bubbles usually spherical? To partly answer this question, assume that
a bubble's surroundings cannot perform work on it. If a spherical bubble maintains a
constant volume, can it become cubic under this circumstance?
(7) A one-metre length of wire is stretched with a constant tension of 10 kN. If the
wire's thermal expansivity is 5 x 10-6 K-1, how much work is done when the wire is
cooled by 20 K? Is the work done on the wire or by the wire?
(8) A block of metal at a pressure of one atmosphere is initially at a temperature of 20
ºC. It is heated reversibly to 32 ºC at constant volume. Calculate the final pressure. The
expansivity  = 5.0 x 10-5 K-1, and the isothermal bulk modulus K = 1.5 x 1011 Nm-2.
For discussion during Tutorial 2 in Week 4 on 26 October, 2011
2TP - Thermal Physics
Department of Physics, University of Surrey
(9) The pressure on a block of iron weighing 100 g is increased isothermally and very
slowly from 0 to 100 MPa. Calculate the work done on the iron. Assume that the density
is unchanged at 7000 kg m-3 and the bulk modulus is constant and equal to 95 GPa.
(10) How much work is done in compressing a van der Waals gas to one-third of its
initial volume under isothermal conditions?
(11) Steel rails (with a length of 10 metres) are placed end to end on a rail line with a gap
between each rail of 1 mm at a temperature of 20 ºC. During the heat wave in August of
2003, the temperature reached 35 ºC. The rails each expanded by 1 mm so that they
were placed in compression. What force is placed on the rail lines when they were
heated up? The linear thermal expansivity  of the steel is 8 x 10-5 K-1, and its elastic
modulus Y is 2 x 1011 Nm-2. The cross-section of the rail is 15 cm by 6 cm.
b
, where b is a
V
constant. Derive an expression for the bulk modulus K of the substance.
(12) (i) The equation of state for a substance is given as P 2T 1 / 3 
(ii) A certain substance discovered on Mars has this equation-of-state: V - bT2 + aP = 0,
where a and b are constants. V, T and P represent the usual state variables. What is the
compressiblity and volume expansivity  of this substance?
(13) Starting with an equation-of-state for one dimension, derive this equation for the
linear expansivity:
1  L 
   ,
L  T 
where T is the temperature,  is the tension, and L is the length of the system.
(14) How much work is required to break a drop of water of radius 3 mm into smaller
aerosol droplets, each with a radius of 5 m? You can assume that the surface tension of
water is constant as the area is changed and is equal to 0.072 N/m at the temperature of
the process.
For discussion during Tutorial 2 in Week 4 on 26 October, 2011