First Law of Thermodynamics

Let $U$ be the internal energy of an ideal gas. Internal energy is the energy that the gas possesses by virtue of the random motions of its constituent molecules. Consider a process by which an infinitesimal amount of heat, $dQ$, is absorbed by the gas, and an infinitesimal amount of work, $dW$, is performed on the gas. According to the first law of thermodynamics, which is a statement of energy conservation that was first explicitly formulated by Rudolf Clausius in 1850,

$$dU= dQ + dW.$$ (5.101)

In reality, $dQ$ cannot be directly measured, and is, instead, inferred to be the difference between the change in the gas's internal energy and the work performed on the gas, both of which can be directly measured, according to the previous equation.

Consider an ideal gas in a cylindrical container of cross-sectional area $A$. Suppose that the top of the container is a movable piston, and that the gas pushes the piston upward a distance $dx$. Now, from the definition of pressure, the gas exerts a force $p\,A$ on the piston. Thus, the gas does work $p\,A\,dx$ on the piston. (See Section 1.3.2.) Hence, the work done on the gas is $dW = - p\,A\,dx= -p\,dV$, where $dV = A \,dx$ is the change in volume of the gas. This is a general result. Hence, Equation (5.101) becomes

$$dU =dQ - p\,dV.$$ (5.102)