Estimate the value of E_{ rot } for the lowest rotational energy state of N _{2}, which has a bond length 0.110 nm.
Strategy If we consider the nitrogen atoms to be point masses (each with mass m) separated by a distance R (see Figure 10.4), then the rotational inertia about an axis passing through the center of the molecule and perpendicular to the line joining the atoms is
I=m\left(\frac{R}{2}\right)^{2}+m\left(\frac{R}{2}\right)^{2}=\frac{m R^{2}}{2}With this numerical estimate of rotational inertia I, the energy levels follow from Equation (10.2).
E_{ rot }=\frac{\hbar^{2} \ell(\ell+1)}{2 I} (10.2)