Entropy is a state function 
, whose differential is expressed as 
.
With reference to eq98, 
for a reversible cycle. Dividing eq98 by ,
)
Substituting the definition of 
from the first law of thermodynamics in eq116:
)
We have shown in an earlier article that 
is an exact differential for an ideal gas. Since the line integral of any exact differential involved in a cyclic process is zero,
)
The function appears to be the differential of a state function. If it is, we must prove that eq118 applies to any working substance, not just an ideal gas. The proof follows the same logic as the proof that eq98 is applicable to any working substance in a Carnot engine. This is accomplished by replacing with and repeating the steps from eq99 through eq105. We call this new state function, entropy, 
:
)
Note that 
in eq119 is the same for both the system and the reservoir (surroundings) for a reversible process. For a closed system undergoing reversible processes, we can substitute eq119 in the differential form of eq24 to give
)
If only pV work is involved,
)
Similarly, we can compute 
for an irreversible process if we can find a reversible process or a combination of reversible processes from point A to point B. Eq121, which is the combination of the first and second laws of thermodynamics, is called the fundamental equation of thermodynamics.
