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# Isothermal Process

Top
 Sub Topics Thermodynamics deals with the conversion of heat in to work or the use of work to produce a cooling or heating effect. In the analysis of practical problems one must determine thermodynamic properties. The most fundamental of the thermodynamic properties are pressure, volume and temperature. States of a thermodynamic system can be changed by interacting with its surrounding through work and heat. When this change occurs in a system, it is said that the system is undergoing a process. A thermodynamic cycle is a sequence of different processes that begins and ends at the same thermodynamic state. The important thermodynamic processes are isothermal, adiabatic, isobaric, isochoric process etc. The definition, properties of isothermal process is described below:

## Isothermal Process Definition

If a system is perfectly conducting to the surroundings and the temperature remains constant through out the process, it is called an isothermal process.

## Work done in Isothermal Process

The expression for work done in isothermal process is given as follows.
From ideal gas equation we know that,
PV=nRT............(1)

We know that, W = PV
So, $\Delta$W=$\int_{v_{1}}^{v_{2}}$PdV.............(2)

From (1) we can write P as,
P =$\frac{nRT}{V}$...............(3)
Substitute (3) in (2)
$\Delta$W = P$\int_{v_{1}}^{v_{2}}$dV

$\Delta$W = nRT$\int_{v_{1}}^{v_{2}}\frac{dV}{V}$

$\Delta$W = nRT ln $\frac{V_{2}}{V_{1}}$
............(4)

Work done is given by,

$\Delta$W = nRT ln $\frac{V_{2}}{V_{1}}$

## Reversible Isothermal Process

Reversible reaction takes place in infinitesimal small steps. Consider a gas reversibly expand from the volume V1 toV2 at constant temperature, the pressure of the gas should reduced from P1 to P2.

Initially, the external pressure is equal to the internal pressure of gas by adjusting the piston. From the above section, the work done is given by,
$\Delta$W = nRT ln $\frac{V_2}{V_1}$............(1)
We know that,
P1V1 = P2V2................(2)

$\frac{P_{1}}{P_{2}}$=$\frac{V_{2}}{V_{1}}$

In terms of pressure, equation (1) becomes
ΔW = nRT ln $\frac{P_{1}}{P_{2}}$...............(3)