Derive mathematical form of First law of Thermodynamics.
Or
Derive the relationship between heat, internal energy and work.

Mathematical form:
The internal energy of the system can be changed in two ways:
(i) by allowing heat to flow into the system or out of the system,
(ii) by work is done on the system or work done by the system.
Let us consider a system which undergoes a change from one state to another. Let the initial internal energy of the system be U₁. Now if a system is supplied q amount of heat, then internal energy of the system increases and becomes U₁ + q.
If work (w) is done on the system, then its internal energy further increases and becomes U₂. The energy U₂ is the energy in the final state.

The above expression is the mathematical form of the first law of thermodynamics. w includes all kinds of work such as pressure-volume work, electrical work etc.
If the work is done in the above process is only pressure-volume work,