**State and path functions and their differentials**

**State of the system**

**State of a thermodynamic system is described by its ****measurable or macroscopic (bulk) properties.**

**â€˘ When the fundamental properties such as ****pressure, volume, temperature, number of moles and composition**** have definite values, the system is said to be in any definite state.**

**â€˘ When there is any change in any one of these properties it is said that the system has undergo a change of state. i.e., ****we need to describe the system by specifying it before and after the change.**

**â€˘ Hence the change of system from initial state to final state is accompanied by the change in state variables. ****i.e., we specify the state of the system by state variables or state functions.**

**State functions / Thermodynamic functions / State Variables**

Those physical quantities whose valuedepend only upon the state of the systemand does not depend upon the path by which this state has been attained are said to be state function.

In other words, the change in the state function accompanying the change in the state of the system depends only upon initial and final states of the system and not on the path by which the change is brought about.

The state function is a property of a thermodynamic system which has a definite value for a particular state of the system.

It is independent of the manner in which the state is reached.For example–Pressure, volume, temperature and energyetc. are state functions.

The change in volume is given by,

\(\Delta V = V_B-V_A\)

where, V_{A}= volume of initial state

and V_{B}= volume of final state

__Differential of State Function__

**State functions follow ****Euler’s reciprocity relation.**** Thus, ****Differential of state function is an ****exact or perfect differential.**

**Exact differential can be ****integrated**** between the appropriate limits.**

** Hence we can write,**

**$$\int_{V_A}^{V_B}dV = V_B – V_A = \Delta V $$**

**Path functionÂ **

path function is the thermodynamic property of a system whose valuedepends on the pathor manner by which the system goes from its initial state to the final state.

Examples of path function:Work, Heat

__Differential of Path Function__

**Path functions ****donot follow ****Euler’s Reciprocity Relation.**** ****Thus, ****Differential of path function is an ****inexact differential****.**

**Inexact differential ****cannot be integrated**** between the appropriate limits. i.e.,**

**$$\int_{q_A}^{q_B}dq \neq q_B – q_A $$**

**$$\int_{W_A}^{W_B}dW \neq W_B – W_A $$**

**Other Important Questions and Points Related to State Functions and Path Functions :**

**Pressure is a state function. Why?**

**because the pressure in a given change of state depends only upon the initial and final ****states of the system and not on the path by which the change is brought about. **

**For example :Â Pressure inside an inflated ****balloon**** does not depends on the path by which it has attained whether through mouth or through a pressure pump. It only depends on the pressure when the balloon was in the initial condition (****deflated)**** and in the final condition (****inflated) ****.**

**Let us consider an example of a pure gas, where the composition is fixed. ****The remaining variables p, V, T are interrelated in the form of a relationship called the equation of state. Thus ****the equation of state**** for 1 mole of a pure gasÂ is pV = RT. where R is the universal gas constant.**

**If any of p, V or T is changed it act as state variable and ***the other two which depends on first one are state functions.*

**Work and Heat are not a state function. Why?**

**because the work done in a given change of state depends upon the path or manner in which the change is brought about.**

**For example, the value of work done in a reversible process is different from that of irreversible process.**

**Similarly, there is ****some change in temperature in system as well as in surrounding ****when heat transfer is taking place. i.e., ****there must be some work done by or work done on the system and work done is a Path Function.**

**In another words, ****heatÂ is defined as energy transferred between the system and the surroundings during some process and the amount that transfers depends on how the process happens. Therefore heat is not a state function.**

**Why work done is a path function and potential energy is a state function ?**

**Consider an another interesting example to differentiate between the path functions and state functions.**

**Let us consider sitting a car on the top of a parking garage.**

**The potential energy of car will be same whether it was lifted by a crane or pushed up by a group of people, or by a helicopter.**

**But the amount of work done or expended to get the car at the top will be different depending on the path chosen.**

**Hence work done depends on the path while the potential energy is independent of the path. **

**How many state variables are present in the equation pV = nRT ?**

**As pressure, volume and temperature are the state variables.**

**While number of moles (n), and gas constant ‘R’ are neither a state function nor a path function.**

**Hence there are 3 state variables in the given equation.**