COMBINED FIRST AND SECOND LAW OF THERMODYNAMICS

Combined 1^st and 2^nd Law of Thermodynamics.

By first law of thermodynamics, ΔQ = dU + ΔW

Since ΔW = pdV

ΔQ = dU + pdV ----------------- 1.

From second law of thermodynamics, (Entropy concept)

ΔQ = T. dS ----------------------- 2.

Put equation 2. in 1.

We get, TdS = dU + pdv ---------------- 3.

We know that, enthalpy h = u + pv

On differentiating, we get

dh = du + pdv + vdp,

From equation 3. , dh = Tds + vdp

Tds = dh - vdp -------------- 4.

The equations 3. and 4. are the thermodynamic equations relating the properties of system.

The following are the relations obtained from the first and second laws.

1. dQ = dE + dW: Holds good for all process, reversible or irreversible and for all systems.

2. dQ = dU + dW: Holds good for any process undergone by a closed system.

3. dQ = dU + pdV: It is good for a closed system , where pdV work is present. This relation true only for quasi-static* process.

4. dQ = TdS: This equation is true only for a reversible processes.

5. TdS = dH –Vdp: This relation hold good for any process, since there is no path function term in the equation. 

6. TdS = dU + pdV: It is good for any reversible or irreversible process, undergone by a closed system. Since the properties in the relation which are independent of the path. 

Note:

* A quasi-static process is a thermodynamic process that happens very slowly  for the system to be in equilibrium. It is reversible.