Chapter 5, Problem 5.1P

Interpretation Introduction

Interpretation:

It is impossible that two adiabatic reversible processes can intersect on $PV\text{}$ diagram. Prove the statement and start with consider false statement.

Concept Introduction:

From the second law of thermodynamics, in order to exchange the heat, you should have two reservoirs at different temperature or all the heat from hot reservoir cannot completely convert to work, some amount of heat from hot reservoir must be exhausted to cold reservoir.

Answer to Problem 5.1P

Hence, it is impossible that two adiabatic reversible processes can intersect on $PV\text{}$ diagram and it violates the second law.

Explanation of Solution

Given information:

It is given that two adiabatic reversible processes do intersect and complete the cycle on $PV\text{}$ diagram with a line representing a reversible, isothermal process.

From the given data, we plot a diagram on $PV\text{}$ diagram involve two adiabatic reversible process intersect at point C and one reversible isothermal process connects both adiabatic reversible process through line AB.

Thus, AB is isothermal process; BC and AC are adiabatic reversible processes. This forms a cycle ABC where area inside the cycle is given as work.

Now, for the heat calculations, the adiabatic reversible processes have no heat transfer, so heat given by both process BC and AC are zero. So, heat transfer takes place only by the isothermal process which has only one single temperature $T_1$ which violates the second law of thermodynamics.

Conclusion

Two thermodynamic curves of same kind (isothermal, adiabatic, polytropic) never intersect each other on $PV\text{}$ diagram. If they intersect from one state to another state without exchange of some energy then they violates both first and second of thermodynamics.