Question

What is mathematical equation of second law of thermodynamics?

Answer 1

To understand why entropy increases and decreases, it is important to recognize that two changes in entropy have to considered at all times. The entropy change of the surroundings and the entropy change of the system itself. Given the entropy change of the universe is equivalent to the sums of the changes in entropy of the system and surroundings:

ΔSuniv=ΔSsys+ΔSsurr=qsysT+qsurrT(1.1)(1.1)ΔSuniv=ΔSsys+ΔSsurr=qsysT+qsurrT

In an isothermal reversible expansion, the heat q absorbed by the system from the surroundings is

qrev=nRTlnV2V1(1.2)(1.2)qrev=nRTln⁡V2V1

Since the heat absorbed by the system is the amount lost by the surroundings, qsys=−qsurrqsys=−qsurr.Therefore, for a truly reversible process, the entropy change is

ΔSuniv=nRTlnV2V1T+−nRTlnV2V1T=0(1.3)(1.3)ΔSuniv=nRTln⁡V2V1T+−nRTln⁡V2V1T=0

If the process is irreversible however, the entropy change is

ΔSuniv=nRTlnV2V1T>0(1.4)(1.4)ΔSuniv=nRTln⁡V2V1T>0

If we put the two equations for ΔSunivΔSunivtogether for both types of processes, we are left with the second law of thermodynamics,

ΔSuniv=ΔSsys+ΔSsurr≥0(1.5)(1.5)ΔSuniv=ΔSsys+ΔSsurr≥0

where ΔSunivΔSuniv equals zero for a truly reversible process and is greater than zero for an irreversible process. In reality, however, truly reversible processes never happen (or will take an infinitely long time to happen), so it is safe to say all thermodynamic processes we encounter everyday are irreversible in the direction they occur.