Why are heat and work not state functions in thermodynamics?

In thermodynamics, heat ($Q$) and work ($W$) are not state functions because their values depend on the specific path taken to transition a system between two states, rather than solely on the initial and final states themselves.12

A state function, such as internal energy ($U$), is defined by the parameters that uniquely determine the system's equilibrium condition. Consequently, any change in a state function ($\Delta U$) is independent of the process used to achieve it. In contrast, heat and work represent energy in transit across the boundary of a system. Because the amount of energy transferred as heat or work can vary depending on the mechanical or thermal process—such as the rate of expansion or the specific sequence of pressure and volume changes—they are considered path-dependent quantities.123

Mathematically, this distinction is expressed through differential equations: the change in a state function is an exact differential ($dU$), whereas the infinitesimal changes in heat ($d^{\prime }Q$) and work ($d^{\prime }W$) are inexact differentials, meaning their integrals cannot be evaluated without specifying the path.41