Question

Clausius inequality: working with a cycic path shown and using his own result, clausius concluded that in an isolated system entropy increases . Can you prove the same as well?

Answer 1

We know the definition of delta S = dq/T

According to the Clausius inequality the dS ≥ dq/T

Work done for reversible process :

We know work done by the system is more in the reversible than irreversible process.

                       Equation of internal energy

                      E = q + w

                Internal energy is state function so its value depending only on final and initial state of the system.
            dE rev = dE irr

And therefore

dq_rev + dWrev = dq_irr + dWirr

lets separates the q and w terms

dq_rev - dq_irr = dw_irr - dw_rev

but we know

dw_irr - dw_rev ≥ 0   …..( system gives work has – sing so when dw_rev is more then that becomes greater than or equal to 0

therefore

dq_rev - dq_irr   ≥ 0

dq_rev ≥ dq_irr    

Lets divide this by T

dq_rev /T   ≥ dq_irr /T

This shows that Delta Srev= dq_rev /T    ≥ 0    (Delta S of irreversible process is always increasing )

And hence it is proved.

d S ≥ dq / T