Prove that internal energy is a state function

Question

Why is it that heat and work are path functions but their sum i.e. internal energy is a state function? How is it even possible?

$ΔU=q+W$

Answer 1

The three laws of thermodynamics (or four, depending on the source) are postulates that scientists have assembled in an attempt to describe the way the universe works. The First Law is a reformulation of the principle of conservation of energy. It says very simply that while energy is indestructible it can be exchanged in various ways. It also implies that a system can exist in different states, each with a set of clearly identifiable properties including a fixed amount of energy. Declaring the state a system is in also stipulates the energy of the system. We can write this mathematically as $U=f(\mathrm{state})$.

One also presumes that if the energy of a system changes, then energy has been transmitted in some measurable form by an identifiable mechanism. In other words, for some process you can always write $$\Delta U = \sum_x \mathrm{energy~in~transit~via~mechanism~}x$$ where the sum goes over all the ways we imagine energy might be transferred. Classical thermodynamics suggests partitioning the energy in transit into two basic categories, work and heat. Heat is transferred when objects are at different temperatures (this forms the basis for the Zeroth Law of thermodynamics). If you insulate a system, heat is no longer a viable way to transfer energy. That leaves all other possible ways of transferring energy, which we call work. Alternately, if all identifiable means to do work are inaccessible, then the system can exchange energy only as heat.

The point is that if the initial and final states are known then $\Delta U$ is fixed, but there might be different ways (paths) that change the energy of the system by that amount. For instance, work can be used to change the temperature of a substance just as putting the substance in contact with an external heat bath can.