What is the mathematical form of 1st law of thermodynamics?

Mathematical formulation of first law of thermodynamics. If heat q is supplied to a system, it may be used up partly to increase the internal energy of the system and partly to produce some mechanical work. If DE is increase in internal energy of the system and w is the work done by the system, then we will have, q = DE + w This equation is the mathematical formulation of first law of thermodynamics. It may be written in a number of other forms as follows; DE = q w. If the changes are very small, it may be written as; dE = dq dw. If the work done is the work of expansion only and the volume increases by a small amount dV against the external pressure P, then we know that dw = PdV Therefore, E = q PdV. From the above equation, it is clear that at constant volume I.e. dV = 0. The change in internal energy is equal to heat absorbed or evolved by the system.

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