Energy of 10 non rigid diatomic molecules at temperature $$T$$ is :

We need to find the total energy of 10 non-rigid diatomic molecules at temperature $$T$$ by applying the equipartition theorem.

According to the equipartition theorem, each degree of freedom contributes $$\frac{1}{2}k_B T$$ of energy per molecule, and the total energy per molecule is therefore $$E = \frac{f}{2} k_B T$$, where $$f$$ is the number of degrees of freedom and $$k_B$$ is Boltzmann's constant.

In a non-rigid diatomic molecule, there are three translational degrees of freedom (motion along the x, y, and z axes), two rotational degrees of freedom (rotation about the two axes perpendicular to the bond axis, since rotation about the bond axis is negligible), and two vibrational degrees of freedom (one for kinetic energy and one for potential energy). Hence, the total number of degrees of freedom is $$f = 3 + 2 + 2 = 7$$.

Substituting $$f = 7$$ into the expression for the energy per molecule gives $$E_{\text{per molecule}} = \frac{7}{2} k_B T$$.

For 10 molecules, this becomes $$E_{\text{total}} = 10 \times \frac{7}{2} k_B T = \frac{70}{2} k_B T = 35 \, k_B T$$.

The correct answer is Option (2): $$35 K_B T$$.