Chapter 29, Problem 7P

To determine

The $N_2$ bond length.

Answer to Problem 7P

The answer is $1.10\times {10}^{-10}m$ .

Explanation of Solution

Given data:

Characteristic rotational energy, $\frac{{\hslash }^{\text{2}}}{2\text{I}}=2.48\times {10}^{-4}\text{eV}$

Radius of spacecraft $=1km$

Formula used:

  $\frac{{\hslash }^{\text{2}}}{2\text{I}}=\frac{{\hslash }^{\text{2}}}{{\text{m}}_{\text{N}}{\text{r}}^{\text{2}}}$

Where,

  $\frac{{\hslash }^{\text{2}}}{2\text{I}}$ is characteristic rotational energy.

  $\hslash$ is reduced Planck’s constant.

  $m$ is mass of atom.

  $r$ is distance.

Calculation:

The bond length can be calculated using: $\frac{{\hslash }^{\text{2}}}{2\text{I}}=\frac{{\hslash }^{\text{2}}}{{\text{m}}_{\text{N}}{\text{r}}^{\text{2}}}$

  $\therefore \frac{{\hslash }^{\text{2}}}{2\text{I}}=\frac{{\left(1.055\times {10}^{-34}\text{J}•s\right)}^2}{14\left(1.66\times {10}^{-27}\text{kg}\right){\text{r}}^{\text{2}}}$

  $\therefore r=\sqrt{\frac{{\left(1.055\times {10}^{-34}\text{J}•s\right)}^2}{14\left(1.66\times {10}^{-27}\text{kg}\right)\left(2.48\times {10}^{-4}\text{eV}\right)}}$

  $\therefore r=1.10\times {10}^{-10}m$

Conclusion: The bond length is $1.10\times {10}^{-10}m$ .