Question 3.18.1: A heat engine is operating in a Carnot cycle: all processes ...

The values of q for each process can be found in Section 3.11.
The engine :
1. For the reversible isothermal expansion process from A to B :

2. For the reversible adiabatic expansion process from B to C :

3. For the reversible isothermal compression process from C to D :

4. For the reversible adiabatic compression process from D to A :

The total change in entropy of the engine is therefore

\Delta S^{total}_{engine} = R\ln\left\lgroup\frac{V_B}{V_A} \right\rgroup+R\ln\left\lgroup\frac{V_D}{V_C} \right\rgroup=0, since \frac{V_B}{V_A}=\frac{V_D}{V_C}    (see Problem 3.7)

For each of the processes, the change in entropy of the surroundings is the negative of that for the engine. Thus, \Delta S^{total}_{surroundings} =0 .

Thus, for the Carnot cycle, \Delta S_{total} =\Delta S_{system} + \Delta S_{surrounds} =0.