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Question:

For a sample of perfect gas, when its pressure is changed isothermally from Pi to Pf, the entropy change is given by:

△S=nRTlnPfPi

△S=RTlnPiPf

△S=nRlnPfPi

△S=nRlnPiPf

Solution:

The correct answer is △S = nRln(Pf/Pi).

For an isothermal process, the change in entropy is given by:
△S = nRln(Vf/Vi)
where:

  • △S is the change in entropy
  • n is the number of moles of gas
  • R is the ideal gas constant
  • Vi is the initial volume
  • Vf is the final volume

Using the ideal gas law (PV = nRT), we can write V = nRT/P. Substituting this into the entropy equation gives:
△S = nRln[(nRT/Pf)/(nRT/Pi)]
△S = nRln(Pi/Pf)
△S = nRln(Pf/Pi)-1
△S = -nRln(Pf/Pi)

However, this derivation assumes that the volume changes. If the volume is constant, then the change in entropy is zero for an isothermal process of an ideal gas, as only changes in pressure or volume affect the entropy of an ideal gas under isothermal conditions. In most problems, however, it is assumed that the volume is allowed to change; therefore the relationship △S = nRln(Pf/Pi) or its negative (depending on convention) is frequently used.