Question

The amount of heat required to raise the temperature of 3 moles of an ideal diatomic gas from 100 degree Celsius to 200 degree Celsius when no work is done is

Answer 1

The heat capacity of anything tells us how much heat is required to raise a certain amount of it by one degree. For a gas we can define a molar heat capacity C - the heat required to increase the temperature of 1 mole of the gas by 1 K.

Q = nCΔT

The value of the heat capacity depends on whether the heat is added at constant volume, constant pressure, etc. Instead of defining a whole set of molar heat capacities, let's focus on CV, the heat capacity at constant volume, and CP, the heat capacity at constant pressure.

Heat Capacity at Constant Volume

Q = nCVΔT

For an ideal gas, applying the First Law of Thermodynamics tells us that heat is also equal to:

Q = ΔEint + W, although W = 0 at constant volume.

For a monatomic ideal gas we showed that ΔEint = (3/2)nRΔT

Comparing our two equations

Q = nCVΔT and Q = (3/2)nRΔT

we see that, for a monatomic ideal gas:

CV = (3/2)R

For diatomic and polyatomic ideal gases we get:

diatomic: CV = (5/2)R

polyatomic: CV = 3R

This is from the extra 2 or 3 contributions to the internal energy from rotations.

Because Q = ΔEint when the volume is constant, the change in internal energy can always be written:

ΔEint = n CV ΔT