Let us now suppose that the same system undergoes the same change in an irreversible manner at a constant temperature T. Entropy being a state function , the increase in entropy of the system will be the same irrespective of the fact whether the change has occurred reversibly or irreversibly.

Thus, ΔS will remain the same as before and represented by the expression,

ΔS_system = q_rev/T …(i)

However, actual amount of heat absorbed by the system from the surroundings will be less, being equal to q_irr

It also follows from here that heat lost by the surroundings will be less than q_rev.However, in view of the infinitely large size of the surroundings, the lost heat by the surroundings is supposed to take place reversibly , i.e. , infinitelyslowly . therefore, mathematically the entropy change of the surroundings is given by the expression,

ΔS_surroundings = – q_irr/T …(ii)

The total entropy change for the combined system and the surroundings ,is,therefore,given by ,

ΔS_system + ΔS _surroundings = q_rev/T – qirr/T

Now, since work done in reversible process is maximum , therefore, we can say that

w_rev > w_irr …(iv)

Also, as internal energy (U) is a state function ,therfore, the value of ΔU remains the same whether the process is caried out in a reversible or an irreversible manner.

As such we can say that,

ΔU= q_rev – w_rev = q_irr – w_irr …(v)

From equation (iv) and (v) we can say that

q_rev > _qirr

or,

q_rev/ T > q_irr/T

or,

q_rev/T – q_irr/T > 0 …(vi)

where ‘T‘ is the temperature at which the change takes place.

From equation (iii) and (vi), we can say that,

ΔS_system + ΔS_surroundings > 0

i.e , Entropy change for the combined system and the surroundings in an irreversible process is greater than zero , In other words , an irreversible process is accompained by a net increase of entropy.

It may, therefore be concluded from above discussion that entropy of the system and its surroundings taken together increase in a thermodynamically irreversiblr process at constant temperature but remains unchanged in a thermodynamically reversible process

Thus, for a thermodynamically reversible process

ΔS_system + ΔS_surroundings = 0 and,

for an thermodynamically irreversible process

ΔS_system + ΔS_surroundings > 0

on combining both the results , we can say that

ΔS_system + ΔS_surroundings ≥  0

where sign ‘equal to‘ refers to a reversible process while the sign ‘greater than‘ refers to an irreversible process

The above generalizations can also be used to distinguish between a thermodynamic reversible and an irreversible process.

Second law of Thermodynamics in term of entropy change – Entropy of the Universe : since all process in nature are occurring spontaneously and are thermodynamically irreversible, it follows, therefore. that entropy of the universe is continously increasuing . This is yet another statement of Second law of Thermodynamics.

Clausius summed up the essentials of first law and second law of thermodynamics as :

The energy of the universe remains constant but entropy of the universe is continously increasing and tends towards a maximum.