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The number of moles of diatomic gas is\(n = 1\;{\rm{mol}}\)

The pressure of diatomic gas at point a is\({P_a} = 2 \times {10^5}\;{\rm{Pa}}\)

The volume of diatomic gas at point a is\({V_a} = 0.010\;{{\rm{m}}^3}\)

The pressure of diatomic gas at point a is\({P_b} = 4 \times {10^5}\;{\rm{Pa}}\)

The volume of diatomic gas at point a is \({V_b} = 0.005\;{{\rm{m}}^3}\)

Whether the process is isothermal compression or not is decided by checking pressure and temperature between two positions. The temperature at point a and b is calculated to check isothermal process.

The temperature at position a is given as:

\({P_a}{V_a} = nR{T_a}\)

Here,\(R\)is the universal gas constant and its value is\(8.31\;{\rm{J}}/{\rm{mol}} \cdot {\rm{K}}\).

Substitute all the values in the above equation.

\(\begin{aligned}\left( {2 \times {{10}^5}\;{\rm{Pa}}} \right)\left( {0.010\;{{\rm{m}}^3}} \right) = \left( {1\;{\rm{mol}}} \right)\left( {8.31\;{\rm{J}}/{\rm{mol}} \cdot {\rm{K}}} \right){T_a}\\{T_a} = 240.67\;{\rm{K}}\end{aligned}\)

The temperature at position b is given as:

\({P_b}{V_b} = nR{T_b}\)

Substitute all the values in the above equation.

\(\begin{aligned}\left( {4 \times {{10}^5}\;{\rm{Pa}}} \right)\left( {0.005\;{{\rm{m}}^3}} \right) = \left( {1\;{\rm{mol}}} \right)\left( {8.31\;{\rm{J}}/{\rm{mol}} \cdot {\rm{K}}} \right){T_b}\\{T_b} = 240.67\;{\rm{K}}\end{aligned}\)

The temperature at positions a and b of process ab are same and pressure for this process is rising from position a to position b so the process ab is an isothermal compression.

Therefore, the process ab is an isothermal compression because the pressure is rising at same temperature.