Question

In high-temperature sources, sodium atoms emit a doublet with an average wavelength of 1139 nm. The transition responsible is from the 4s to 3p state. Set up a spreadsheet to calculate the ratio of the number of excited atoms in the 4s state to the number in the ground 3s state over the temperature range from an acetylene-oxygen flame (3000°C) to the hottest part of an inductively coupled plasma source 8750°C).

Short answer

The ratio of sodium atom at the acetylene-oxygen flame,${3000}^{\circ }\mathrm{C}=0.063128709$

The ratio of sodium atom at an inductively coupled plasma source,data-custom-editor="chemistry" $8750°\mathrm{C}=0.739341631$

Step-by-step solution

Given information

In high-temperature sources, sodium atoms emit a doublet with an average wavelength of 1139 nm. The transition responsible is from the 4s to 3p state.

Explanation

Boltzmann equation is used for the calculation of the ratio. This equation tells that how much an atom or ion is populated as a function of temperature. This equation is given as-

$\frac{N_j}{N_0}=\frac{g_j}{g_0}exp\left(\frac{-E_j}{KT}\right)$

Calculation of Energy

And the calculation of the energy of atom and ion is done by the following formula-

$E_j=\frac{hc}{\lambda }$

The wavelength for the Na atom when the transition of an atom occurs from 4s to 3p = $1139\mathrm{nm}$, Planck’s constant = $6.62607\times {10}^{-34}\mathrm{J}.\mathrm{s}$, $\mathrm{c}=3\times {10}^8\mathrm{m}/\mathrm{s}$

localid="1645797748501" $$\begin{gathered} E_j=\frac{6.6207\times {10}^{-34}\mathrm{J}.\mathrm{s}\times 3\times {10}^8\mathrm{m}/\mathrm{s}}{1139\times {10}^{-9}\mathrm{m}} \\ {E}_j=1.744\times {10}^{-19}\mathrm{J} \end{gathered}$$

For 4s is 2 and for 3p is 6. Thus, the ratio can be given as:

$\left(\frac{g_j}{g_0}\right)=\frac{6}{2}=3$

Data on spreadsheet

Calculation of the ratio of excited state 4s to the ground state 3s for sodium atom is done by plotting a spreadsheet. Therefore, from the spreadsheet-

$\left(\frac{g_j}{g_0}\right)$	3
	1.74E-19
k	1.38E-19
T (℃)	T(K)	$\frac{N_j}{N_0}$
3000	3273	0.063128709
3250	3523	0.083027718
3500	3773	0.105305209
3750	4023	0.129672246
4000	4273	0.155835534
4250	4523	0.183511052
4500	4773	0.212432389
4750	5023	0.242355284
5000	5273	0.273059559
5250	5523	0.304349331
5500	5773	0.336052158
5750	6023	0.368017546
6000	6273	0.400115144
6250	6523	0.43223281
6500	6773	0.464274679
6750	7023	0.496159323
7000	7273	0.527818022
7250	7523	0.559193195
7500	7773	0.590236979
7750	8023	0.620909954
8000	8273	0.651180024
8250	8523	0.681021421
8500	8773	0.71041383
8750	9023	0.739341631
9000	9273	0.767793232

Conclusion

The ratio of sodium atoms at ${3000}^{\circ }\mathrm{C}=0.063128709$

The ratio of sodium atom atdata-custom-editor="chemistry" $8750°\mathrm{C}=0.739341631$