We have already seen that the internal energy of a gas can be changed both by heat \(\Delta Q\) and by the performance of mechanical work \(p \Delta V\). In both cases, the kinetic energy and thus the velocity of the gas particles increases. Therefore, one can generally write:
$$
\Delta U = \Delta Q + p \Delta V
$$
This formulation, which is essentially nothing other than the law of conservation of energy for thermodynamic systems, strictly speaking only holds for uniform changes of the three quantities.In the general case, infinitesimally small changes must be considered:
$$
\mathrm{d}U = \mathrm{d}Q + \mathrm{d}W
$$
This is the standard form of the First Law of Thermodynamics, which can be found in the literature.
When the system performs work, \(\mathrm{d}W\) is negative, because the internal energy must then decrease. If work is done on the system, \(\mathrm{d}W\) is correspondingly greater than 0.
The First Law, just like the conservation of energy, cannot be derived from more fundamental principles. It reflects empirical experience and the fact that no one has ever succeeded in building a perpetual motion machine that produces more energy than is put into it. Such a system would, in analogy to the First Law, be called a perpetual motion machine of the first kind.