In a diatomic gas, the molar specific heat capacity at constant volume (Cv) and molar specific heat capacity at constant pressure (Cp) are related by the equation:
Cp - Cv = R
where R is the molar gas constant.
In the given scenario, the gas does work of Q/4 when a heat of Q is supplied to it. According to the first law of thermodynamics, the change in internal energy (ΔU) of a system is equal to the heat supplied (Q) minus the work done (W):
ΔU = Q - W
For a diatomic gas, the change in internal energy can be expressed as:
ΔU = CvΔT
where ΔT is the change in temperature.
Since the gas does work of Q/4, we can write:
W = Q/4
Therefore, the change in internal energy becomes:
ΔU = Q - (Q/4) = 3Q/4
Since ΔU = CvΔT, we have:
CvΔT = 3Q/4
The molar specific heat capacity at constant volume (Cv) is defined as the amount of heat required to raise the temperature of one mole of a substance by one degree Celsius at constant volume. Therefore, we can write:
Q = CvΔT
Substituting this into the earlier equation, we get:
CvΔT = 3(CvΔT)/4
Simplifying the equation, we find:
ΔT = 3ΔT/4
This equation implies that ΔT is equal to zero. However, this contradicts the fact that heat (Q) is being supplied to the gas, which should result in a temperature increase.
Hence, there seems to be an inconsistency in the given scenario. Please double-check the information provided or provide additional details if necessary.