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Firstly, a state function is a property whose value does not depend on the path taken to reach that specific value. In other words, state functions are variables that define a system. Heat is not a state function because heat is amount of thermal energy that enters or leaves the system during a process. Thus, heat is a change in energy, not an amount of contained energy in the system.
Heat is not a state function because it does not depend on the initial and final conditions of a system. For example, enthalpy which describes the change in the heat of the system depends on the difference between the final and initial state of the system. Therefore, as long as the final and initial states remain the same, the difference between the two will always stay the same independent of the path taken to reach the final state. Heat, however, describes a quantity rather than a difference. The current heat of the system does not depend on the initial and final state of the system, it does not equal to the difference between the two. By calculating the difference between the final and initial states, we cannot get the heat of the system now. We can only get the heat of the system by either adding the change to the initial state or by measuring it directly. By definition heat is the transfer of energy that results from a difference in temperature, which is a state function.
Enthalpy is a state function since it depends on both the final and initial conditions of the system. Heat would not be a state function because heat is a measurement of change in energy and does not depend on the final and initial conditions of the system.
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