Question

The vibrational frequency of the ICI molecule is$\mathbf{1}\mathbf{.15}\mathbf{\times }{\mathbf{10}}^{\mathbf{13}}\mathbf{}{\mathbf{\text{s}}}^{\mathbf{-}\mathbf{1}}$ . For every million $\mathbf{(}\mathbf{1}\mathbf{.00}\mathbf{\times }{\mathbf{10}}^{\mathbf{6}}\mathbf{)}$molecules in the ground vibrational state, how many will be in the first excited vibrational state at a temperature of$\mathbf{300}\mathbf{}\mathbf{\text{K}}$ ?

Short answer

The molecules in the first excited vibrational state is$16\times {10}^4$.

Step-by-step solution

Given information

The vibrational state of a million molecules and temperature is given as:

$\begin{array}{c}\nu =1.15\times {10}^{13}\\ {\text{s}}^{-1}T=300\text{K}\end{array}$

Concept of Maxwell–Boltzmann distribution

The Maxwell–Boltzmann distribution is concerned with the energy distribution between identical but distinct particles. It denotes the likelihood of the distribution of states in a system with varying energies.

The so-called Maxwell distribution law of molecular velocities is a specific instance.

Calculate the relative probability

The relative probability of finding molecules in the excited stateand in the ground stateis given by:

$\begin{array}{l}\frac{P\left(n\right)}{P\left(0\right)}=\exp \left(\frac{-[{\epsilon }_n-{\epsilon }_0]}{k_{B}T}\right)\\ \frac{P\left(n\right)}{P\left(0\right)}=\exp \left(\frac{-h\nu }{k_{B}T}\right)\end{array}$

Or in other words,$\frac{P\left(n\right)}{P\left(0\right)}=\exp \left(\frac{-h\nu }{k_{B}T}\right)$.

Where$k_B$is the Boltzmann constant.$=1.38\times {10}^{-23}{\text{J}}^{-1}{\text{K}}^{-1}$

So, to determine the number of molecules there is a need to calculate the ratio of molecules in the ground and in the first vibrational state.

Calculate  

Substitute the above values in the equation.

$\begin{array}{c}h=6.625\times {10}^{-34}\text{J}/\text{s}\\ h\nu =6.625\times {10}^{-34}\times 1.15\times {10}^{13}\\ h\nu =7.62\times {10}^{-21}\text{J}\end{array}$

Then calculate the number of molecules in the first excited vibrational state.

$\begin{array}{l}N=0.16\times 1\times {10}^6\\ N=16\times {10}^4\end{array}$