Explain the internal energy as a state function.

(N/A) Internal energy is a state function that represents the total energy of a system,including chemical,electrical,mechanical,and other forms of energy. In thermodynamics,it is denoted by $U$. The internal energy of a system changes when:
$\Rightarrow$ Heat passes into or out of the system.
$\Rightarrow$ Work is done on or by the system.
$\Rightarrow$ Matter enters or leaves the system.
To demonstrate that internal energy is a state function,consider an adiabatic system (a system insulated from its surroundings,where $q = 0$).
$1$. Let the initial state of the system be $A$ with temperature $T_{A}$ and internal energy $U_{A}$.
$2$. We can change the state of the system to state $B$ (with temperature $T_{B}$ and internal energy $U_{B}$) in two different ways:
- Way $I$: Perform mechanical work (e.g.,$1 \ kJ$) by churning water with paddles.
- Way $II$: Perform an equal amount of electrical work (e.g.,$1 \ kJ$) using an immersion rod.
In both cases,the final temperature $T_{B}$ is found to be the same. Since the change in internal energy $\Delta U = U_{B} - U_{A}$ depends only on the initial and final states ($A$ and $B$) and not on the path taken to reach the state,internal energy is a state function.