Isothermal process

In an isothermal process, the system's pressure and volume can vary while the temperature remains constant. Heat exchange between the system and its surroundings is permitted during this process. Achieving an isothermal state typically requires either supplying heat to the substance or extracting heat from it. It's important to note that for a process to be truly isothermal, it must proceed very slowly.

1.0Definition of Isothermal Process 

In the realm of thermodynamics, an isothermal process is identified by the unique property of sustaining a consistent temperature within the system from start to finish, irrespective of any changes occurring within it. This means that any changes in pressure, volume, or other variables occur while the temperature remains unchanged. In an isothermal process, heat exchange between the system and its surroundings is allowed to maintain the constant temperature.

2.0Equation of Isothermal Process

                 P V = constant (Where, T is constant)

3.0Work Done in Isothermal Process

Consider µ moles of an ideal gas, enclosed in a cylinder, at absolute temperature T, fitted with a frictionless piston. Suppose that gas undergoes an isothermal expansion from the initial state (P₁, V₁) to the final state

(P₂, V₂) Work done:

$\mathrm{W}={\int }_{V_1}^{V_2}\mathrm{PdV}\therefore \mathrm{PV}=\mu \mathrm{RT}\text{ then }P=\frac{\mu RT}{V}\therefore \mathrm{W}={\int }_{{\mathrm{V}}_1}^{{\mathrm{V}}_2}\frac{\mu \mathrm{RT}}{\mathrm{V}}d\mathrm{V}=\mu \mathrm{RT}{\int }_{{\mathrm{V}}_1}^{{\mathrm{V}}_2}\frac{\mathrm{dV}}{\mathrm{V}}=\mu \mathrm{RT}{\left[{\log }_e\mathrm{V}\right]}_{{\mathrm{V}}_1}^{{\mathrm{V}}_2}=\mu \mathrm{RT}\left[{\log }_e{\mathrm{V}}_2-{\log }_{\mathrm{e}}{\mathrm{V}}_1\right]=\mu \mathrm{RT}{\log }_{\mathrm{e}}\left[\frac{{\mathrm{V}}_2}{{\mathrm{V}}_1}\right]$

$\Rightarrow \mathrm{W}=2.303\mu \mathrm{RT}{\log }_{10}\left[\frac{{\mathrm{P}}_1}{{\mathrm{P}}_2}\right]\left[\because {\mathrm{P}}_1{\mathrm{V}}_1={\mathrm{P}}_2{\mathrm{V}}_2\right]$

Note :- By the First Law of Thermodynamics , There is no change in temperature and internal energy of the system depends on temperature only 

So ΔU = 0, Q = 2.303 µRT log₁₀ $\left[\frac{{\mathrm{v}}_2}{{\mathrm{V}}_1}\right]$

It is clear that Whole of the heat energy supplied to the system is utilized by the system in doing external work. There is no change in the internal energy of the system.

4.0Graph of Isothermal Process

For an isothermal process, PV = constant

Differentiating, PdV + VdP = 0 

⇒  VdP = – PdV 

$\Rightarrow \frac{dP}{dV}=-\frac{P}{V}$ 

Slope of isothermal curve, ${\left[\frac{\mathrm{dP}}{\mathrm{dV}}\right]}_{\text{isothermal }}=-\frac{\mathrm{P}}{\mathrm{V}}$

5.0Points to be remember in Isothermal Process

• The product of the pressure and volume of a given mass of a perfect gas remains constant in an isothermal process. 
• Boyle's law is obeyed in an isothermal process.
• The representation of pressure against volume for a specific mass of gas at a constant temperature is termed an isotherm or isothermal curve, denoting the behavior of the gas under these conditions.
• The curves drawn for the same gas at different temperatures are mutually parallel and do not cut each other.
• If two isotherms intersect each other at a single point we get same value of P and V at intersection point. PV = µRT₁ for temperature T₁ and PV = µRT₂ for temperature T₂. It means T₁ = T₂ is not possible. 
• An ideal gas enclosed in a conducting cylinder fitted with a conducting piston. 
• Let the gas be allowed to expand very–very slowly. This shall cause a very slow cooling of the gas, but heat will be conducted into the cylinder from the surrounding. Hence the temperature of the gas remains constant. If the gas is compressed very–very slowly, heat will be produced, but this heat will be conducted to the surroundings and the temperature of the gas shall remain constant.
• The temperature of a substance remains constant during melting. So, the melting process is an isothermal process.
• Boiling is an isothermal process, when a liquid boils, its temperature remains constant.

If sudden changes are executed in a vessel of infinite conductivity then they will be isothermal.