# Estimate the relative intensities of absorption given the split in energy levels

Estimate the relative intensities at $$\pu{25 °C}$$ of absorption originated in the ground state and the first excited state when the energy levels involved are separated by:

(a) $$\pu{10000 cm^-1};$$
(b) $$\pu{1000 cm^-1};$$
(c) $$\pu{10 cm^-1}.$$

Here, the given numbers are the wave numbers of the EM radiation.

I have a problem understanding the term “relative intensities”. What does it imply? My professor has told me to calculate it using the population of states using the formula:

$$\frac{N'}{N} = \exp\left(\frac{-\Delta E}{k_\mathrm B T}\right)$$

But my problem is that the term $$N'/N$$ is the ratio of the number of particles in excited state and the ground state. How does it relate to relative intensities?

I'm fairly new to spectroscopy so I might have used few terms in the wrong fashion.

• The relative intensities are proportional to the number of particules. So try to calculate N'/N for the three values of energy levels. You will immediately obtain the result your teacher would like you to get. – Maurice Sep 4 '20 at 14:59
• @Maurice Thanks for help. Can you give some text where this concept is described in detail? Hope you don't mind. – Optimus Sep 4 '20 at 15:04
• You don't need a textbook. You have the formula. Only thing to do is transform the unit $cm^{-1}$ in frequency unit (Hertz), then in energy unit (Joule), and divide the result by $kT$, where $k$ is Boltzmann constant. Hopefully you know the corresponding numerical values. – Maurice Sep 4 '20 at 15:18
• @Optimus IF truly needed, see textbooks about physical chemistry, because Boltzmann constant and distribution (statistical thermodynamics) / Boltzmann statistics. Nuclear Magnetic Resonance spectrosccopy (NMR) and optical pumping of LASERs seek to go beyond what heat taken alone would offer (population inversion). Note however, this detail of understanding may be required only if you study e.g., chemistry, but is presented to you typically in or even past year three. – Buttonwood Sep 4 '20 at 15:27
• Do you know the equation $E = h\nu = \frac{hc}{\lambda}$? – Mathew Mahindaratne Sep 4 '20 at 15:55