The increase in internal energy of an ideal gas is equal to the heat added to the gas for the processes that occur at constant volume (b) and adiabatic (d) processes.
a) Constant temperature: In this process, the internal energy of the gas remains constant because the temperature doesn't change. Any heat added to the gas is used to do work, resulting in an increase in the gas's external energy rather than its internal energy.
b) Constant volume: In this process, the volume of the gas remains constant, so no work is done by the gas. Therefore, any heat added to the gas directly increases its internal energy.
c) Constant pressure: In this process, the pressure of the gas remains constant. Work is done by the gas as it expands against the external pressure, and this work affects the change in internal energy. Therefore, the increase in internal energy is not equal to the heat added to the gas.
d) Adiabatic: In an adiabatic process, no heat is transferred between the gas and its surroundings. Any change in internal energy is solely due to the work done on or by the gas. Therefore, the increase in internal energy is equal to the work done, and there is no heat added or subtracted.
So, the answer is (b) constant volume and (d) adiabatic processes.