# Related to the Second Law

It is impossible to cross two adiabatic curves. Could someone better justify me why? Is it something related to the Second Law?

• That the so-called (reversible) adiabatic surfaces do not have a common point is almost equivalent to the "2nd" law that is to the existence of the entropy function. Each such surface can be uniquely numbered in a monotonic fashion numerically corresponding to the entropy of the system. A (reversible) adiabatic surface = isentropic surface, (reversible) adiabatic process = isentropic process. – hyportnex Nov 12 '19 at 17:09
• Exactly what do you mean by “cross”. Intersect at a point? – Bob D Nov 12 '19 at 17:11
• @BobD Intersect at a point! – Meulu Elisson Nov 12 '19 at 17:14
• How can I analyze graphically? – Meulu Elisson Nov 12 '19 at 17:19
• See my answer below – Bob D Nov 12 '19 at 17:51

Two adiabatic curves can intersect, but one of the adiabatic processes has to be irreversible. If both processes are carried out quasi-statically, then the irreversible process involves friction (a process can be quasi-static but irreversible due to friction). To illustrate, let's assume an ideal gas undergoes two adiabatic processes, one irreversible due to friction and one reversible.

For any adiabatic process, applying the first law

$$\Delta U=-W$$

And since for any ideal gas, any process

$$\Delta U=C_{V}(T_{i}-T_{f})$$

Although in an adiabatic process there is no heat transfer between the system and the surroundings, if the process involves friction that will result in an increase in internal energy as if heat were transferred to the gas. Since for an ideal gas the internal energy depends only on temperature, as indicated above, the final temperature for a given final volume will be higher for the process with friction and thus the decrease in internal energy will be less, and less work performed for the adiabatic process with friction.

The diagram below shows a reversible and irreversible (due to friction) adiabatic process for an ideal gas, both starting with the same initial conditions. That is the one point where the two processes intersect, or "cross".

Hope this helps.