State and Path Functions - Definition, Difference Between State Functions and Path Functions, Practice Problems and FAQs

A full-service airline where two people are travelling from say point A to Point B. Let’s consider Point A is at the foot of the hill, while Point B is on the top of the hill.

While one person took a ropeway and reaches the top in one direct single climb, the other took a toy train that makes a gradual climb around the hill in a circular motion with loops and bends and reaches the top. Distance travelled by both the people is different but both achieved the same elevation.

The same destination for both of them but they took different routes to reach Point B from Point A. If we see, both of them have taken the same vertical displacement to reach the top of the hill. However, the distances they both have covered are different. So, here the question arises, out of distance and displacement which is the one that depends on the path taken? 

Well! Distance travelled by both of them is different, so distance depends on the path but displacement or elevation here will be the same between Point A and Point B for both the persons. So, displacement is rather state dependent. 

Now, in thermodynamics, there are two types of properties or functions, one is a state function the other is the path function. Let’s study these two functions in detail.

TABLE OF CONTENTS

• System
• Surroundings
• Boundary
• Universe
• Types of Systems
• State of the system
• Difference between State and Path functions
• List of state functions
• Practice Problems
• Frequently Asked Questions - FAQs

System

A system in the thermodynamic term refers to that part of the universe in which thermodynamic observations are made.

Surroundings

Apart from the system, everything else in the universe is considered as surroundings.

Boundary

A boundary is a physical barrier that separates the system from its surroundings. A system is segregated from its surroundings by a real or imaginary wall. 

Tea in the vessel is defined as a system and everything else as surroundings. Here, the wall of the container which is separating the system and surroundings which exists, in reality, is known as a real boundary.

When a small portion of tea (within the entire volume of the container marked in white colour) is defined as a system, the boundaries of this system are imaginary as it does not exist in reality and it can be changed if some amount of tea is poured out. 

Universe 

The system and its surroundings make up the universe.

Types of Systems

• Open System: There is an interchange of energy and matter between the system and its surroundings in an open system.
  
  

• Closed System: In a closed system, there is no exchange of matter. However, an exchange of energy is possible between the system and the surroundings.
  
  

• Isolated System: There is no exchange of energy or matter in an isolated system between the system and the surrounding. Thermos flask is an example of an isolated system where the exchange of matter and energy will not take place.
  
  

To summarise this:

Systems can also be classified as follows:

• Ideal system: The term ideal system refers to a hypothetical mechanical system in which energy and power are not lost or dissipated through friction, deformation, wear, or other inefficiencies. They exist at high temperatures and low pressures. This does not exist in reality.

• Real system: A real-world system can be thought of as an example of a real system. They exist at high pressure and low temperature. Generally, all open systems where the transfer of energy and matter both takes place are considered real systems.

State of the System 

When the macroscopic properties of a system have definite values, the system is said to be in a definite state. Whenever there is a change in any one of the macroscopic properties, the system is said to be changed into a different state. As a result, the system's state is determined by its macroscopic properties. The state of the system is the condition in which the system is present. The state is defined by measuring the observable properties of the system like pressure, volume, temperature etc. 

• State functions: These are the physical quantities whose values depend only on the state of the system and do not depend upon the path by which this state has been attained. A few examples of state functions are pressure, volume, temperature, etc.

• Path functions: These are the physical quantities that are dependent on the path by which the system has achieved a particular state. Heat and work are examples of path functions.

Here we can observe the path dependent nature of work and the path-independent nature of potential energy. The man is going to the third floor using a lift and the woman by stairs, the work done while reaching the third floor is different for both but since they both arrive at the third floor. Whereas, the potential energy is the same for the man and the woman because both are gaining the same height.

Take another illustration of work as a path function is given in the figure below. Here, the work done by the athletes while running around the track will depend on the path followed by them, i.e., if the athlete chooses to run through another path (i.e., the trajectory of the athlete changes), then the work done by the athlete during running will also change.

Take one more illustration to show the state function as given in the figure below. There is a change in volume as a consequence of the expansion and compression when the plunger of the syringe is pushed and pulled. This change in volume, ΔV, will depend only on the initial and final volumes and does not depend upon how we press the syringe.

Difference Between State Function And Path Function

State functions, as previously described, are attributes whose values are independent of the path taken to reach that function or value. Path functions are the functions that rely on the path followed to get to a given value.

List of State Functions

Pressure: Pressure is the average force imposed on the container walls by the constituent molecules per unit area. Pressure is a state function since it is independent of the molecules’ path.

Temperature: Temperature is the degree of coldness and hotness of a body. Temperature is a state function because it measures an attribute of a system’s state. 

Volume:

The quantity of physical space filled by a material is measured in volume, which is independent of the path taken. As a result, the volume is a function of the state.

Mass:

The amount of substance in an object is quantified by its mass, which is commonly expressed in grams (g) or kilograms(kg). The quantity of matter is measured by mass, which is a state function that is independent of its location in the universe or the gravitational force acting on it.

Internal energy:

It is defined as the total amount of energy connected with molecular movements.

The internal energy of ideal gases is solely a function of temperature, whereas the internal energy of real gases is a function of temperature, pressure, and volume (temperature and volume being the dominant quantities, and the effect of pressure being negligible). It can thus be seen that because internal energy is dependent on state functions such as P, T, and V, the internal energy is also a state function.

Gibb’s free energy:

Gibb's free energy of the system is the enthalpy of the system at any location minus the product of the temperature times the entropy of the system.

G = H – TS

The system's Gibbs free energy is a state function since it is represented in terms of thermodynamic parameters that are state functions.

Entropy:

Entropy is a measure of the randomness of the system. It is unique to the current state of the system, hence entropy is a state function. 

Recommended Video Link: https://youtu.be/Xq0EipqWJJY 

Practice Problems

Q 1. A system wherein the exchange of energy and matter takes place with the environment is called: 

a. Open System 
b. Closed System
c. Isolated System
d. None of these 

Answer: (A)
In thermodynamics, a closed system can exchange energy (as heat or work) but no matter, with its surroundings. An isolated system cannot exchange both energy and matter with the surroundings, while an open system can exchange both energy and matter with the surroundings.

Q 2. The system where only energy can be exchanged with the surroundings and not the matter is called:

a. Open System 
b. Closed System
c. Isolated System
d. None of these 

Answer: (B)
In thermodynamics, a closed system can exchange energy (as heat or work) but no matter, with its surroundings. An isolated system cannot exchange any heat, work, or matter with the surroundings, while an open system can exchange both i.e. energy and matter.

Q 3. Which of the following is a state function?

A) q w
B) q +w
C) q2w2
D) q -w

Answer: (B)
According to the first law of thermodynamics;

 ΔU=q+w

As we know that ΔU (change in internal energy) is a state function, thus we can say that q+w is a state function.

Q 4. Which is not a state function?

a. Enthalpy 
b. Entropy
c. Work
d. Internal Energy

Answer: (C)
Heat and work are not state functions, they are path functions. Work cannot be a state function because it is proportional to the distance travelled by an object, which depends on the path used to get from one state to the next. 

Q 5. Which of the following is a state function and an extensive property at the same time?

a. Internal energy 
b. Pressure
c. Molar heat capacity 
d. Temperature

Answer: (A)
Internal energy only depends upon the initial and final state of the system, not on the path followed. As a result, the internal energy function is a state function.

The amount of matter has an impact on internal energy. The internal energy increases as the amount of the substance increases, causing more collisions. So, the internal energy is an extensive property.

Frequently Asked Questions – FAQ

Q 1. Is altitude a state function?
Answer: It doesn’t matter how you got to the top of the mountain, the change in altitude will be the same. This is an example of a state function, which is a property whose value is independent of the path used to get there.

Q 2. Why entropy is a state function?
Answer: Entropy is a state function since it depends only on the initial and the final states of the system. It does not depend on the path, whichever path it takes to reach a final state from the initial state, the entropy value remains the same for both the states and the change in entropy will be constant between two states.

Q 3. Why energy is a state function but heat and work are not even though both are one of the form of energy? 

Answer: Actually energy has two forms:

1. Energy in storage: Energy which can be stored. Examples: chemical energy, internal energy, kinetic energy, potential energy, pressure energy etc. All energies in storage are state functions.

2. Energy in transition: Energy which can not be stored but is in the transition state. Example: heat, work, electrical energy etc. All energies in transition are not state functions, they are path functions. 

Hence energy is a path function but heat and work are not.

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