Question

# The value $\gamma$ of a triatomic gas (linear arrangement) molecules is\begin{align} & A.\dfrac{5}{3} \\ & B.\dfrac{7}{5} \\ & C.\dfrac{8}{6} \\ & D.\dfrac{9}{7} \\ \end{align}

Hint: Triatomic gas consists of molecules of these gases which have three atoms. As a result these are known as triatomic having atomicity equal to three. Some of the most general examples of triatomic gases is carbon dioxide, water vapour, nitrous oxide, ozone etc.

The ratio of the specific heat at constant pressure to the specific heat at constant volume is the $\gamma$
Therefore the specified term can be expressed in an equation like,
$\gamma =\dfrac{{{C}_{P}}}{{{C}_{V}}}$
This is a factor happening in the adiabatic engine processes and results in determining the speed of sound in a gas. Now let us discuss degrees of freedom. A degree of freedom is an independent physical parameter in the formal description of the state in a physical system. The set of all states of a system is called the system's phase space, and the degrees of freedom of the system is the dimensions of the phase space. As we all know, the degrees of freedom for the triatomic gas is given as 7 in number.
As per the relation,
$\gamma =1+\dfrac{2}{f}$
Here $f$ is the number of degrees of freedom and $\gamma$ is the constant.
Therefore after substituting the given values in the equation will give,
$\gamma =1+\dfrac{2}{7}=\dfrac{9}{7}$

So, the correct answer is “Option D”.

Note:
Molecular degrees of freedom is described as the number of ways a molecule in the gas phase can move, rotate, or vibrate in space. Three types of degrees of freedom are persisting, those being translational, rotational, and vibrational. This ratio $\gamma =1.66$ for an ideal monatomic gas and $\gamma =1.4$ for air, which is predominantly called as a diatomic gas.