Question

The molar heat capacity of an ideal gas
A. Is zero for an adiabatic process
B. Is infinite for an isothermal process
C. Depends only on the nature of the gas for a process in which either volume or pressure is constant.
D. Is equal to the product of the molecular weight and specific heat capacity for any process.

Answer 1

Hint:The amount of heat energy required to raise the temperature of one mole of a substance is known as its molar heat capacity. It is denoted by the symbol \[{c_n}\].

Formula used:
Mathematically, the formula for the molar heat capacity of an ideal gas is as written below:
\[Q = n{c_n}\Delta T\]…….(i)
Or
\[{c_n} = \dfrac{Q}{{n\Delta T}}\]……(ii)
Where \[Q\] is the heat energy required, \[n\] is the amount in moles and \[\Delta T\] represents the change in temperature.

Complete step by step solution:
(A) For an adiabatic process, \[Q = 0\]
Substituting the value in equation (ii),
\[{c_n} = \dfrac{0}{{n\Delta T}}\]
\[{c_n} = 0\]
Therefore, the molar heat capacity for an adiabatic process is zero. Therefore, option A is correct.

(B) For an isothermal process, the temperature is constant. Therefore, the change in temperature will be zero that is \[\Delta T = 0\]
Substituting the value in equation (ii), we get
\[{c_n} = \dfrac{Q}{0}\]
\[{c_n} = \infty \]
The molar heat capacity for an isothermal process is infinite. Therefore, option B is correct.

(C) For an ideal gas, when the volume is constant, the molar heat capacity is given as:
\[{c_v} = \dfrac{R}{{\gamma - 1}}\]
Where\[R\]is the ideal gas constant and \[\gamma \] depends on the nature of gas.
\[{c_v} = \gamma \]
When pressure is constant, the molar heat capacity is given as:
\[{c_p} = \dfrac{{\gamma R}}{{\gamma - 1}}\]
\[{c_p} = \gamma \]
As \[\gamma \] depends on the nature of the gas. Therefore, molar heat capacity for an ideal gas depends only on the nature of the gas for a process in which either volume or pressure is constant.
Therefore, option C is correct.

(D) Molar heat capacity is given by the equation
\[Q = mc'\Delta T\]……(iii)
where\[m\] is the mass, \[c'\] is the specific heat capacity and \[\Delta T\] is the change in temperature
Comparing (i) and (iii),
\[nc\Delta T = mc'\Delta T\]
\[c = \dfrac{{mc'}}{n}\]
\[\dfrac{m}{n}\] is the molecular weight and can be written as \[M\].
The above equation becomes,
\[c = Mc'\]
Therefore, the molar heat capacity of an ideal gas is equal to the product of the molecular weight and specific heat capacity for any process. Therefore, option D is correct.

Hence, options A, B, C and D are correct.

Note: When pressure is constant, the heat absorbed increases the internal energy as well as helps in doing work. But when the volume is constant, it only increases the internal energy. As a result, the specific heat is more when pressure is constant.