State Functions:
1. Definition and Examples
- State functions are properties that depend only on the current state of a system and are independent of the path taken to reach that state.
- These functions are often represented by capital letters, such as temperature (T), pressure (P), volume (V), internal energy (U), enthalpy (H), Gibbs free energy (G), and entropy (S).

2. Examples of State Functions:
- Temperature (T): The measure of the average kinetic energy of particles in a system.
- Pressure (P): The force exerted per unit area.
- Volume (V): The amount of space occupied by a system.
- Internal Energy (U): The sum of the kinetic and potential energies of all particles in a system.
- Enthalpy (H): The total heat content of a system at constant pressure.
- Gibbs Free Energy (G): A measure of the maximum amount of useful work that can be obtained from a system at constant temperature and pressure.
- Entropy (S): A measure of the disorder or randomness in a system.

Path Functions:
1. Definition and Examples:
- Path functions are properties that depend on the path taken to reach a particular state of a system.
- These functions are not solely determined by the initial and final states of the system but also by the process or pathway followed.

2. Examples of Path Functions:
- Work (w): The energy transferred to or from a system as a result of a force acting on it through a displacement.
- Heat (q): The energy transferred to or from a system due to a difference in temperature.
- Change in Internal Energy (ΔU): The difference between the initial and final internal energies of a system.

Explanation:
State functions are independent of the path taken and only depend on the current state of the system. For example, the temperature of a system remains the same regardless of how it reached that temperature. Similarly, pressure and volume are also state functions as they only depend on the current state of the system.

On the other hand, path functions are dependent on the path taken to reach a particular state. Work and heat are examples of path functions. The amount of work done on a system or the heat transferred to a system depends on the specific process or pathway followed.

It is important to note that while state functions can be determined by measuring the properties of the system, path functions cannot be directly measured. Instead, they are calculated by analyzing the changes in state functions between the initial and final states of the system.

Examples of state functions include density, internal energy, enthalpy, entropy. Such a relation cannot be written for path functions, especially since these cannot be defined for the limiting states. Path functions depend on the route taken between two states. Two examples of path functions are heat and work.