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Equivalently, perpetual motion machine of the first kind i.e a
"hypothetical machine which, once activated, would continue to function
and produce work" indefinitely with no input of energy" is impossible .

2.6 Application of the First Thermodynamic Law to Isoprocesses

Taking into account the relation dQ  dU  dA it is possible to

obtain:
dQ dU dA
C    (2.6.1)
dT dT dT

hence the heat capacity also depends on the type of the thermodynamic
process.
1) Isochoric process;  constV  dV  0, since  pA dV  0. Hence,

dQ  dU (2.6.2)

All heat given to the system is spent on an alteration in the internal energy
of the system.

i i
For ideal gas U   RT  dU   RdT , therefore according to the
2 2
dQ
definition of molar heat capacityC  (compare with formula 2.4.2)
  dT
we obtain expression of molar heat capacity of ideal gas at constant
volume

i
 RdT
dQ dU 2 i
C     R (2.6.3)
V
  dT   dT dT 2

2) Isobaric process: p  const, since dQ  dU  dA. So dA  p  dV
and
take into account the ideal gas equation of state pV  RT follows

pdV  RdT  dA (2.6.4)

Then

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