Why is energy a state function but work is a path function?

State function

In thermodynamic equilibrium, a?state function,?function of the state, or?point function?for a?thermodynamic system is a?mathematical function relating several?state variables or state quantities (that describe?the equilibrium states of a system) that depend only on the current equilibrium thermodynamic state of the system.? A state function describes the equilibrium states of a system, thus also describing the type of system. A state variable is typically a state function so the determination of other state variable values at an equilibrium state also determines the value of the state variable as the state function at that state.?

Explanation

The energy of a system, such as water, depends on its current state, which is determined by factors like temperature, pressure, and composition. In the case of water, its energy, specifically its enthalpy, can vary depending on whether it is in a liquid or gaseous state. When water is in the liquid state, at ambient pressure, its enthalpy is H1 = 419 KJ/kg [about]. However, when it undergoes a phase change and becomes saturated steam, its enthalpy H2 becomes much higher than H1 KJ/kg, H2 = 2260 KJ/kg. This difference in enthalpy illustrates how the energy of water depends on its state. This concept applies to other systems as well. Whether it is a gas, solid, or liquid, the energy of a system is influenced by its state, and changes in energy can occur when the system undergoes transformations or phase changes. Therefore, energy is indeed a state function, as it is determined by the current characteristics of the system and not by the specific pathway it took to reach that state.

Path function

Path functions depend on the path taken to reach one state from another. Different routes give different quantities. Examples of path functions include?work and heat. In contrast to path functions,?state functions are independent of the path taken.?

Gases follow a?polytropic process?which is a?thermodynamic process that obeys the relation PV^n = Constant. The polytropic process equation describes expansion and compression processes which include heat transfer and work transfer.

Here?P?is the?pressure,?V?is?volume,?n?is the?polytropic index, and?C?is a constant. Therefore, the work done by gas is multiple of pressure P and V^n where n depends on the path of work or heat transfer.

Some specific values of?n?correspond to particular cases:

for an?isobaric process , n =0

for an?isochoric process, n = + infinity

In addition, when the ideal gas law applies:

for an?isothermal process, n = 1

for an?isentropic process, n =y

Where?y?is the ratio of the heat capacity at constant pressure (Cp) to heat capacity at constant volume (Cv).

Therefore, work or heat transfers between two points have 0 to infinite path through it can be accomplished.

Example

The polytropic equation PV^n = constant is a general equation for gases, and the value of the polytropic index (n) determines the specific path and work done.

To demonstrate that work is a path function, let's consider two different paths between the same initial and final states of a gas. For convenience, let's assume the initial state is given by P1, V1, and the final state is given by P2, V2.

Path 1: A → B

Let's assume this path is an isobaric process, where the value of n is zero (n = 0) in the polytropic equation. In this case, the work done (W1) can be calculated as:

W1 = ∫(P1 to P2) PdV = ∫(V1 to V2) (constant/V^0) dV = constant ∫(V1 to V2) dV = constant (V2 - V1).

Path 2: A → C → B

Let's assume this path is an isochoric process, where the value of n is infinity (n = ∞) in the polytropic equation. In this case, the work done (W2) can be calculated as: W2 = ∫(P1 to P3) PdV + ∫(P3 to P2) PdV = ∫(V1 to V3) (constant/V^∞) dV + ∫(V3 to V2) (constant/V^∞) dV = 0 + 0 = 0.

In the isochoric process work done is indeed zero.

Comparing W1 and W2, we see that they are different. Therefore, the work done is not the same for different paths. Hence, we can conclude that work is indeed a path function for gases. It depends on the specific path taken and the value of the polytropic index (n) in the polytropic equation PV^n = constant.