Question

The molar heat capacity for 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: Change in heat,$\Delta Q$ is given by the equation
$\Delta Q = nC\Delta T$
Where $n$ is the number of moles, $C$ is the molar heat capacity and $T$ is the temperature.
The adiabatic process is a process in which there is no heat exchange.
So, $\Delta Q = 0$
The isothermal process is a process in which the change in temperature is zero.
That is, $\Delta T = 0$
For constant pressure molar heat capacity is denoted as ${C_P}$
It is given by the equation
${C_p} = \dfrac{{\gamma R}}{{\gamma - 1}}$
Where $\gamma $ is the specific heat ratio which depends on the nature of gas and $R$ is the universal gas constant.
For constant volume molar heat capacity is denoted as ${C_V}$
It is given by the equation
${C_V} = \dfrac{R}{{\gamma - 1}}$
Molar heat capacity is also given by the equation,
$\Delta Q = mc\Delta T$
Where $m$ is the mass and $c$ is the specific heat capacity.

Complete step by step solution:
We know that $\Delta Q$ is given by the equation
$\Delta Q = nC\Delta T$ ……. (1)
Therefore,
$\dfrac{{\Delta Q}}{{n\Delta T}} = C$ …… (2)
Where $n$ is the number of moles, $C$ is the molar heat capacity and $T$ is the temperature.
The adiabatic process is a process in which there is no heat exchange.
So, $\Delta Q = 0$
Substitute this in equation (2). Then we get
$ \Rightarrow C = \dfrac{0}{{n\Delta T}} = 0$
So option A is correct.
The isothermal process is a process in which the change in temperature is zero.
That is, $\Delta T = 0$
Substitute this in equation (2). Then we get,
$
  C = \dfrac{{\Delta Q}}{{n\left( 0 \right)}} \\
   = \infty \\
 $
So, option B is correct
For constant pressure molar heat capacity is denoted as ${C_P}$
It is given by the equation
${C_p} = \dfrac{{\gamma R}}{{\gamma - 1}}$
Where $\gamma $ is the specific heat ratio which depends on the nature of gas and $R$ is the universal gas constant.
For constant volume molar heat capacity is denoted as ${C_V}$
It is given by the equation
${C_V} = \dfrac{R}{{\gamma - 1}}$
Therefore, the 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.
So, option C is correct
Molar heat capacity is also given by the equation,
$\Delta Q = mc\Delta T$ …… (3)
Where $m$ is the mass and $c$ is the specific heat capacity.
Now compare equation (1) and (3). We get,
$nC\Delta T = mc\Delta T$
$
   \Rightarrow nC = mc \\
   \Rightarrow C = \dfrac{{mc}}{n} \\
 $
We know $\dfrac{m}{n}$ is the molecular weight $M$. Therefore,
$C = Mc$
Thus, option D is also correct.

Hence, all options are correct.

Note:
To find whether the molar heat capacity for an ideal gas is equal to the product of the molecular weight and specific heat capacity we can use dimensional analysis. Unit of specific heat, $c$ is $Jk{g^{ - 1}}mo{l^{ - 1}}$ and molecular weight, $M$ has unit $kg$ so there product will have unit $Jmo{l^{ - 1}}$ which is same as the unit of molar heat capacity $C$ . Hence, it is correct.