Chapter 7, Problem 7.13P

Interpretation Introduction

Interpretation:

The number of revolutions taken by hydrogen atoms in given water molecule and the energy required to excite the rotations of the molecule has to be calculated.

Concept Introduction:

The revolutions made per second are obtained by dividing the angular speed value by $\text{2π}$.

The expression that relates the angular speed, inertia and angular momentum is given as follows,

$\begin{array}{l}\text{Revolutions}\text{=}\frac{\text{ω}}{\text{2π}}\\ =\frac{{\text{J}}_{\text{z}}}{\text{2πI}}\\ =\frac{\hslash }{\text{2πI}}=\frac{\text{h}}{{\left(\text{2π}\right)}^{\text{2}}\text{I}}\\ \boxed{\begin{array}{c}\text{here,ω = angular}\text{speed}\\ \text{I = Moment}\text{of}\text{inertia}\\ \text{h = Planck’s constant }\\ {\text{J}}_{\text{z}}\text{= angular}\text{momentum}\end{array}}\end{array}$

The energies are quantized for particle in sphere and are generally expressed as follows,

$\begin{array}{c}\text{E}=l\left(l+1\right)\frac{{\hslash }^2}{2m{r}^2}\\ l=0,1,2,\mathrm{...}\\ m_l=0,\underset{\_}{+}1,\underset{\_}{+}2,\mathrm{...}\\ here,l=orbitalangularmomentum\text{quantum number}\\ m=mass\\ r=spheresurfaceradius\\ \hslash =\frac{h}{2\pi },hisPlanck\text{'}s\text{constant}\end{array}$