Internal energy is a state function. Explain.

Internal energy is indeed a state function in thermodynamics. This means that its value depends only on the current state of a system and not on the path the system took to reach that state. This is in contrast to path functions, which do depend on the specific path taken.

Here's a more detailed explanation of why internal energy is a state function:

Definition of Internal Energy: Internal energy (U) is a measure of the total energy of a system due to the kinetic and potential energies of its particles (atoms and molecules). It includes the energy associated with the motion of particles (kinetic energy) and the energy associated with their positions relative to each other (potential energy).

Independence of Path: Internal energy depends solely on the current state of the system, which is defined by certain macroscopic properties like temperature, pressure, volume, and the number of particles. It does not matter how the system reached that state or what processes occurred along the way. Whether the system underwent a complex series of transformations or followed a simple, direct path, the internal energy at the final state will be the same, provided that the final state is the same.

Mathematical Expression: Mathematically, the change in internal energy (ΔU) between two states is given by the first law of thermodynamics:

ΔU = Q - W

Where:

ΔU is the change in internal energy.
Q is the heat added to or removed from the system.
W is the work done on or by the system.
The key point is that the change in internal energy depends on the initial and final states of the system, and it is independent of the specific processes (heat transfer or work) that occurred during the transformation.

In contrast, path-dependent functions like work and heat are not solely determined by the initial and final states of the system; they also depend on the specific path taken during the process. For example, the work done on a gas in a piston-cylinder system depends on the path of the piston, while the change in internal energy only depends on the initial and final states of the gas, not how the piston moved.

In summary, internal energy is a state function because it depends only on the current state of the system and not on the process or path taken to reach that state. This characteristic makes it a fundamental concept in the field of thermodynamics.