Question

Molecule of CO at moderate temperature have energy

• A. $\frac{7}{2} k_{B} T$
• B. $\frac{5}{2} k_{B} T$
• C. $\frac{3}{2} k_{B} T$
• D. $\frac{1}{2} k_{B} T$

Answer 1

Answer: (A)

Molecule of CO at moderate temperature possess vibrational modes, i.e. it oscillates like one-dimensional oscillator.
$\therefore$ Vibrational energy $=\frac{1}{2} m\left(\frac{d y}{d t}\right)^{2}+\frac{1}{2} k y^{2}$
As, each vibrational mode contributes two square terms, so this mode will contribute $2\left(\frac{1}{2} k_{B} T\right)$ energy. CO molecule has 3 translational and 2 rotational modes and each of them contributes $\frac{1}{2} k_{B} T$ to the energy.
$\therefore$ Total energy,
$E=\underbrace{3\left(\frac{1}{2} k_{B} T\right)}+\underbrace{2\left(\frac{1}{2} k_{B} T\right)}+\underbrace{2\left(\frac{1}{2} k_{B} T\right)}$
Translational mode Rotational mode Vibrational mode
$=\frac{7}{2} k_{B} T$