Relation Between Heat and Entropy

Consider the heating of a working substance by a reversible process as shown whose base represents the entropy and the vertical ordinate represents the absolute temperature.

Now Consider any point A on the curve 1-2. At this point, let a small quantity of heat (&Q) be supplied to the working substance, which will increase the entropy by dS. Let the absolute temperature at this instant be T. Then according to the definition of entropy,

&Q = TdS

From Fig we see that the terms TdS represents the area under the curve during this change of entropy. 

From Equations we get

ds = &Q/T

The total change in entropy may be obtained by integration the above expression from state 1 to state 2.

then we found

TdS = dU + pdv