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 $=\frac{1}{2}m{\left(\frac{dy}{dt}\right)}^2+\frac{1}{2}k{y}^2$
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
$=\frac{7}{2}{k}_{B}T$