Question

A molecule of gas has six degrees of freedom. Then, the molar specific heat of the gas at constant volume is
a. ${\displaystyle \frac{R}{3}}$
b. $R$
c. ${\displaystyle \frac{3R}{2}}$
d. $2R$
e. $3R$

Answer 1

Hint: solve by using the relation between $C_v$ (molar specific heat at constant volume) and $f$ ( degree of freedom of molecule).

Complete Step by step answer:
Degree of freedom of a molecule refers to the number of ways a molecule in the gas phase may move, rotate, or vibrate in space. Here the degree of freedom of molecule is given
$f=6$

Specific heat of a gas at constant value is given by
$C_v={\displaystyle \frac{f}{2}}\times R$
Where, $R=$universal gas constant
$$\begin{gathered} C_v={\displaystyle \frac{6}{2}}R \\ {C}_v=3R \end{gathered}$$

Hence, (e) is the correct option

Additional information:
- The specific heat capacity of gas depends on the conditions under which it is measured.
- The specific heat at constant volume is defined as the quantity of heat required to raise the temperature of one kilogram of the gas by one kelvin if the volume of the gas remains constant.
- The specific heat capacity at constant pressure is defined as the quantity of heat required to raise the temperature of one kilogram of the gas by one kelvin if the pressure of the gas remains constant.
- The specific heat at constant pressure is always greater than the specific heat at constant volume.
- Molar heat capacity is defined as the heat required to raise the temperature of one mole of gas by one kelvin. If this is done keeping the volume constant then it is known as $C_v$. If this is done keeping the pressure of the gas constant then it is known as $C_P$ .
- The relation between $C_p$ and $C_v$ is given by the Mayor’s formula
$C_p-C_v=R$ , Where, $R$ is the universal gas constant.

Note: ‘Molar specific heat’ should not be confused with ‘specific heat’. Molar specific heat is derived for one mole of gas while specific heat is derived for one kilogram of gas.