# Solutions 10

### Q2:

The number of degrees of freedom a molecule has is 3N, where N is the number of atoms. Three are translational and for non-linear molecules, three are rotational, leaving 3N-6 vibrational modes. For linear molecules there are only two rotational degrees of freedom, leaving 3N-5 vibrational modes.

 Molecule Vibrational Modes $$NH_3$$ 6 $$C_6H_6$$ 30 $$C_{10}H_8$$ 48 $$CH_4$$, 9 $$C_2H_2$$ 7

### Q3:

The one mode of CO stretch is IR active. All but the symmetric stretch of $$CO_2$$ are IR active. The one mode of HCl is also active.

### Q4:

The vibrational frequency is given by $$\omega = \sqrt(\frac{k}{\mu})$$, where $$\mu = \frac{m_1 m_2 }{m_1 + m_2}$$. The masses are provided.

### Q5:

The vibrational energy is given by $$E_\nu = \hbar \omega (\nu + \frac{1}{2})$$.

### Q6:

The corresponding wave number of $$D_2 O$$ compared with $$H_2 O$$ is smaller due to the increased reduced mass, resulting in a lower energy read.