In thermodynamics, work is defined as a mode of energy transfer between a system and its surroundings that occurs when a force acts through a distance. Work is observed only during a process and is not a property stored within the system.
When a thermodynamic system interacts with its surroundings, energy may cross the system boundary either in the form of heat due to a temperature difference or in the form of work due to mechanical or other generalized forces. Once the transfer is complete, the system retains only a change in internal energy and not the work itself.
Work as a Path Function
The amount of work done during a thermodynamic process depends on the path followed between the initial and final states. For this reason, work is classified as a path function and not a state function.
If a system changes from one state to another through different paths, the work done along each path may be different even though the initial and final states are the same. Consequently, the differential of work is written as an inexact differential and represented by the symbol δW.
Sign Convention for Work
To apply thermodynamic equations consistently, a sign convention is adopted for work. When work is done by the system on the surroundings, it is taken as positive and represented by W > 0. When work is done on the system by the surroundings, it is taken as negative and represented by W < 0.
This sign convention is used throughout classical thermodynamics and ensures consistency with the mathematical form of the First Law of Thermodynamics.
Pressure–Volume Work
The most common form of work encountered in thermodynamics is pressure–volume work, which occurs when a system undergoes expansion or compression against an external pressure. This type of work is especially important in the study of gases.
Consider a gas enclosed in a cylinder fitted with a frictionless piston. If the gas expands and pushes the piston outward against an external pressure, the gas does work on the surroundings. If the gas is compressed, work is done on the gas by the surroundings.
Mathematical Expression for Pressure–Volume Work
For an infinitesimal change in volume dV against an external pressure Pext, the infinitesimal work done is given by
\[\delta w = P_{ext}\,dV\]
For a finite change in volume from V1 to V2, the work done is obtained by integration and is given by
\[w = \int_{V_1}^{V_2} P_{ext}\,dV\]
Reversible and Irreversible Work
In a reversible process, the external pressure differs infinitesimally from the internal pressure of the system at every stage, and the work done is maximum. In this case, the work done is given by the integral of pressure with respect to volume along a reversible path.
In an irreversible process, the system expands or is compressed against a finite external pressure, and the work done is less than that obtained in a reversible process between the same initial and final states.
Other Types of Work
In addition to pressure–volume work, other forms of work may be encountered in thermodynamic systems. These include electrical work, surface work, magnetic work, and gravitational work. In each case, work arises due to the action of a generalized force acting through a corresponding displacement.
Although the detailed treatment of these forms of work may vary, they all represent modes of energy transfer and are treated consistently within the framework of the First Law of Thermodynamics.
Work in the First Law of Thermodynamics
According to the First Law of Thermodynamics, the change in internal energy of a system is equal to the heat supplied to the system minus the work done by the system.
\[\Delta U = q – w\]
This equation shows that work plays a central role in determining how energy is distributed between a system and its surroundings during a thermodynamic process.
With the concept of work clearly established, we are now prepared to study pressure–volume work in greater detail through specific thermodynamic processes.
Important Questions and Examples
Short Answer Questions
Q1. Define work in thermodynamics.
Answer: In thermodynamics, work is defined as the mode of energy transfer between a system and its surroundings that occurs when a force acts through a distance.
Q2. Why is work considered a path function?
Answer: Work is considered a path function because its value depends on the path followed during a thermodynamic process and not solely on the initial and final states of the system.
Q3. State the sign convention for work.
Answer: Work done by the system on the surroundings is taken as positive, whereas work done on the system by the surroundings is taken as negative.
Long Answer / Theory Questions
Q4. Explain pressure–volume work with a suitable example.
Answer: Pressure–volume work arises when a system undergoes expansion or compression against an external pressure. For example, when a gas expands in a cylinder fitted with a movable piston, it pushes the piston outward and performs work on the surroundings. Conversely, during compression, work is done on the gas by the surroundings.
Q5. Distinguish between reversible and irreversible work.
Answer: In a reversible process, the external pressure differs infinitesimally from the internal pressure of the system at every stage, and the work done is maximum. In an irreversible process, the system expands or is compressed against a finite external pressure, and the work done is less than that in a reversible process between the same initial and final states.
Numerical Examples
Example 1. Calculate the work done when a gas expands from 5 L to 10 L against a constant external pressure of 1 atm.
Solution: The work done in an irreversible expansion against constant external pressure is given by
\[w = P_{ext}(V_2 – V_1)\]
Here, \(P_{ext} = 1\ \text{atm}\), \(V_1 = 5\ \text{L}\), and \(V_2 = 10\ \text{L}\).
\[w = 1 \times (10 – 5) = 5\ \text{L atm}\]
Converting into joules using \(1\ \text{L atm} = 101.3\ \text{J}\),
\[w = 5 \times 101.3 = 506.5\ \text{J}\]
Example 2. One mole of an ideal gas undergoes reversible isothermal expansion at 300 K from 10 L to 20 L. Calculate the work done.
Solution: The work done in a reversible isothermal expansion is given by
\[w = nRT \ln \frac{V_2}{V_1}\]
Here, \(n = 1\ \text{mol}\), \(R = 8.314\ \text{J mol}^{-1}\text{K}^{-1}\), \(T = 300\ \text{K}\), \(V_1 = 10\ \text{L}\), and \(V_2 = 20\ \text{L}\).
\[w = 1 \times 8.314 \times 300 \times \ln \frac{20}{10}\]
\[w = 2494.2 \times \ln 2 = 2494.2 \times 0.693\]
\[w = 1728\ \text{J (approximately)}\]
This example shows that the work done in a reversible process is greater than that in an irreversible process for the same initial and final states.
Exam-Oriented Practice Questions
1. Define work and explain why it is a path function.
2. Derive the expression for pressure–volume work.
3. Compare reversible and irreversible work with suitable diagrams.
4. Calculate work done during expansion of a gas under different thermodynamic conditions.