**TdS Relations**

Consider the first law:

dQ - dU + dW (1)

For a reversible process:

dQ = TdS & dW = pdV

TdS = dU + pdV (2)

But

H = U +PdV

So that

dH = dU + pdV + Vdp (3)

Substituting 3 into 2 we get:

Tds = dH - Vdp (4)

*Note:*

Equations 2 and 4 were derived for a reversible process and so they can be integrated for a reversible process only. But they deal with properties ie. They only depend on end state so the change in properties during a given change of state are the same for an irreversible process as for a reversible process. Therefore they can be used for irreversible process **as well.**

## Entropy property diagrams

Temperature- entropy or T-S diagrams are very useful for visualising changes of state.

*Notes:*

1. Area under a curve 1-2 in a reversible process corresponds to heat transfer

2. Isentropic processes are vertical straight lines

3. In two phase region constant pressure lines are horizontal

4. The efficiency of a reversible cycle is given by the ratio of the area enclosed by the curve to the area beneath the upper part of the curve. For example in the case of the carnot cycle:

**h-S or Mollier diagram**

Widely used in steam power plant calculations, because enthalpy appears in SFEE and because of importance of isentropic processes for comparison with practical adiabatic expansion or compression.

1. Constant pressure and temperature lines coincide in the 2 phase region

2. Constant temperature lines tend to the horizontal in the superheated region as the steam behaviour approaches perfect gas behaviour

3. The shape of a constant pressure line from Tds = dh Vdp is:

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First law

Second law

Thermodynamics