In an isothermal process, the temperature of the system remains constant. Since the kinetic energy of gas molecules is directly proportional to the temperature, increasing the volume of an ideal gas without increasing the kinetic energy requires careful manipulation of the system.
One way to achieve this is by performing work on the gas. According to the ideal gas law, the volume of a gas is inversely proportional to the pressure when the temperature and the number of moles of gas remain constant:
PV = nRT
where P is the pressure, V is the volume, n is the number of moles, R is the ideal gas constant, and T is the temperature.
By reducing the pressure on the gas while keeping the temperature constant, the volume will increase. This can be accomplished by an external force pushing a piston, allowing the gas to expand against a lower pressure.
To summarize, by reducing the pressure on an ideal gas while maintaining a constant temperature, the volume can be increased without increasing the kinetic energy of the gas molecules. This is accomplished by performing work on the gas, allowing it to expand against a lower pressure.