# Workdone in Thermodynamics

In thermodynamics, work is generally defined as the force multiplied by the distance. If the displacement of body under the force F is ds, the work done will be,
δW = Fds

The symbol δW stands for the small amount of work and also the inexactness of the function. Several things should be noted in the definition of work.

(a) Work appears only at the boundary of the system

(b) Work appears only during a change in state

(c) Work is manifested by an effect in the surroundings

(d) Work is an algebraic quantity. It is positive if the work has been produced in the surroundings. It is negative if the work has been destroyed in the surroundings.

(e) In SI system of units, work is expressed in Joule or Kilojoule 1J = 1 Nm.

### Types of work :

(a) Gravitational Work: The work is said to be done when a body is raised through a certain height against the gravitational field. Suppose a body of mass m is raised through a height h against the gravitational field. Then the magnitude of the gravitational work is mgh.

(b) Electrical work: This type of work is said to be done when a charged body moves from one potential region into another. If the charge is expressed in coulombs and the potential difference in volts, then the electrical work is given by QV.

(c) Mechanical Work: work associated with change in volume of a system against an external pressure is referred to as the mechanical or pressure-volume work.

Now we shall discuss mechanical work in detail.
Work has been done by the system if a weight has been raised in the surrounding, work has been done on the system if a weight has been lowered.

### Work in reversible process :

(a) Expansion of a gas

(b) Suppose n moles of a perfect gas is enclosed in a cylinder by a frictionless piston. The whole cylinder is kept in large constant temperature bath at T°K. Any change that would occur to the system would be isothermal.

Suppose area of cross section of cylinder = a sq. cm

Pressure of the piston = P

Distance through which gas expands = dl cm

Then force (F) = P × a

Work done by the gas = F × dl  ⇒ P × a × dl

∴ a × dl = dV

∴ W = P . dV

Let the gas expand from initial volume V1 to the final volume V2, then the total work done

$\large W = \int_{V_1}^{V_2} P dV$

### Work done in isothermal reversible expansion of an ideal gas

The small amount of work done, δw when the gas expands through, a small volume dV, against the external pressure, P is given by

δw = – PdV

∴ Total work done when the gas expands from initial volume V1 to final volume V2 will be

$\large W = – \int_{V_1}^{V_2} P dV$

Ideal gas equation , P V = n R T

i.e, P = nRT/V

Hence , $\large W = – \int_{V_1}^{V_2} \frac{n R T}{V} dV$
[∵ T = constant]

∴ $\large W = – n R T \; ln\frac{V_2}{V_1}$

$\large W = – 2.303 n R T \;log\frac{V_2}{V_1}$

$\large W = – 2.303 n R T \; log\frac{P_1}{P_2}$