(Solved): 3. The rotational kinetic energy of a molecule is given by Erot=I2/2=L2/2I where L is the ang ...

3. The rotational kinetic energy of a molecule is given by $${\mathrm{E}}_{\mathrm{rot}}=\mathrm{I}{?}^2/2={\mathrm{L}}^2/2\mathrm{I}$$ where $$\mathrm{L}$$ is the angular momentum, given by $$[1(1+1){]}^{1/2}?$$, and $$\mathrm{I}$$ is the moment of inertia, given by $$I=$$ $${\mathrm{m}}_1{\mathrm{r}}_1^2+{\mathrm{m}}_2{\mathrm{r}}_2^2({\mathrm{m}}_1$$ and $${\mathrm{m}}_2$$ are the masses of the atoms and $${\mathrm{r}}_1$$ and $${\mathrm{r}}_2$$ are the distances to the center of mass). (a) Show that I can be written as $$I=?r^2$$, where $$?$$ is the reduced mass. ( 3 pts) (b) The carbon and oxygen atoms in a $$\mathrm{CO}$$ molecule are separated by about $$0.12\mathrm{nm}$$. Calculate the moments of inertia for $${}^{12}\mathrm{CO}$$ and $${}^{13}\mathrm{CO}$$. (Note: the atomic masses for $${}^{12}\mathrm{C}$$, $${}^{13}\mathrm{C}$$, and $${}^{16}\mathrm{O}$$ are $$12.000\mathrm{u},13.003\mathrm{u}$$, and $$15.995\mathrm{u}).(5\mathrm{pts})$$ (c) Calculate the wavelength of a photon emitted when $${}^{12}\mathrm{CO}$$ transitions from state $$\mathrm{l}=$$ 3 to $$l=2$$. In What part of the electromagnetic spectrum is this wavelength? ( 3 pts) (d) Calculate the wavelength of a photon emitted when $${}^{13}\mathrm{CO}$$ transitions from state $$\mathrm{l}=$$ 3 to $$l=2$$. How can astronomers distinguish between emission lines of different isotopomers in the ISM? (3 pts)