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Question: (Molar specific heat capacity at constant pressure) – (Molar specific heat capacity at constant volu...

(Molar specific heat capacity at constant pressure) – (Molar specific heat capacity at constant volume) is equal to
(A) R
(B) P
(C) V
(D) T

Explanation

Solution

To understand and solve this question, we will have to first understand the molar specific heat capacity at constant pressure and at constant volume. We will also use the relationship between them to finally get our solution.

Complete answer:
First let us understand what molar specific heat capacity is at constant pressure and at constant volume.
Molar specific heat capacity at constant pressure is defined as the amount of heat energy required to raise the temperature of one mole of a substance through 1K1K or 1∘C1^\circ C at constant pressure. It is denoted by CP{C_P} .
By definition, CP=dqPdT{C_P} = \dfrac{{d{q_P}}}{{dT}}
Since, qP=H{q_P} = H
Therefore, CP=dHdT{C_P} = \dfrac{{dH}}{{dT}}
Molar specific heat capacity at constant volume is defined as the amount of heat energy required to raise the temperature of one mole of a substance through 1K1K or 1∘C1^\circ C at constant volume. It is denoted by CV{C_V} .
By definition, CV=dqVdT{C_V} = \dfrac{{d{q_V}}}{{dT}}
Since, qV=U{q_V} = U
Therefore, CV=dUdT{C_V} = \dfrac{{dU}}{{dT}}
Now let’s see the relationship between molar specific heat capacity at constant pressure and at constant volume.
From the equation of enthalpy, H=U+PVH = U + PV
But, in case of one mole of ideal gas, PV=RTPV = RT
Therefore, H=U+RTH = U + RT
On differentiating the both sides with respect to temperature, we get,
dHdT=dUdT+d(RT)dT\dfrac{{dH}}{{dT}} = \dfrac{{dU}}{{dT}} + \dfrac{{d(RT)}}{{dT}}
But we know that, CP=dHdT{C_P} = \dfrac{{dH}}{{dT}} and CV=dUdT{C_V} = \dfrac{{dU}}{{dT}}
Therefore,
CP=CV+R.dTdT{C_P} = {C_V} + R.\dfrac{{dT}}{{dT}}
CP=CV+R{C_P} = {C_V} + R
Or we can say, CP−CV=R{C_P} - {C_V} = R
Thus, the difference between the molar heat capacities at constant volume and pressure always equals RR , the universal gas constant.
Hence, option A is correct.

Note:
In exams, you need not to memorize the whole thing. You can just remember the relationship between molar specific heat capacity at constant volume and at constant pressure for one mole, that is, CP=CV+R{C_P} = {C_V} + R . You can also derive the relation between the two by simply using the enthalpy equation.