Derive the various mathematical statements of the first law.

Explanation

Mathematical statement of the First law of Thermodynamics is `DeltaU=q+w`
Case 1: For a cyclic process involving isothermal expansion of an ideal gas
`DeltaU=0`
`therefore` q=-w
In other words, during a cyclic process, the amount of heat absorbed by the system is equal work done by the system.
Case 2: For an isochoric process (no change in volume) there is no work of expansion
`DeltaV=0`
w=0
`DeltaU=q_v`
In other words, during isochoric process, the amount of heat supplied to the system is converted to its internal energy.
Case 3: For an adiabatic process there is no change in heat i.e.q=0. Hence
q=0
`DeltaU=w`
In other words, in an adiabatic process, the decrease in internal energy is exactly equal to the work done by the system on its surroundings.
Case 4 : For an isobaric process. There is no change in the pressure. Remains constant. Hence
`DeltaU=q+w`
`DeltaU=q-PDeltaV`
In other words, in an isobaric process a part of heat absorbed by the system.is used for PV expansion work and the remaining is added to the internal energy of the system