Using the free energy of a monatomic ideal gas, derive the ideal gas law.
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Universal gas constant, R = 8.3145 J mol-1 K-1
1. The Helmholtz function of a certain gas is given by
v – nb)T²
an?
F(T,V) = –nRT|1+ ln|
пф
V
where n is the number of moles of gas, T is the temperatures of the gas, V is the volume of
the gas, and a, b, and are constants.
(a) Derive the equation of state, the entropy, and the internal energy of the gas.
(b) Suppose that 100 moles of the gas expands from 2 m³ to 5 m³ at 280 K and that the
constants have the values a = 0.364 J m³ mol¬2 , b = 4.27 x 10-5 m³ mol¬1, and
$ = 1.09 x 10-5 m³ K³/² mol¯1. Calculate:
(i) aU.
(ii) the maximum energy that can be made available for work due to the process.
(iii) the speed of sound in the gas after the expansion if the ratio of specific heat
capacities for the gas is y = 1.28 and the density of the gas is 0.881 kg m-3 in the
final state.
(c) Using the free energy of a monatomic ideal gas, derive the ideal gas law.
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