Convergence limits

Background

In sub-topic 12.1 Electrons in atoms underneath 'Applications and skills' it states "Calculation of the value of the first ionization energy from spectral data which gives the wavelength or frequency of the convergence limit". It is not that easy for teachers to find a good source of spectral data which gives the convergence limit. For example an internet search for the emission spectrum of sodium gives several images that all appear different and seem to depend upon the pressure under which the spectrum was taken. The example below is from The difference between.net. 

Whilst most show the splitting of the two intense yellow lines it is almost impossible to determine where the convergence limit occurs.

Probably the best source for the data is from The National Institute of Science and Technology (NIST). If you click on one of the elements in the periodic table, e.g. sodium, you will see that it gives the ionization energy for sodium in its ground state (written as 1s22s22p63s 2S1/2) as 41449.451 cm-1. For chemists this is a strange unit for ionization energy. What it actually refers to is the wavenumber (i.e. 1/λ) of the relevant convergence limit. If you wish to use it to give to students then it would be more normal to convert it into a wavelength, i.e. 2.413 x 10-7 m or 241 nm. To calculate the first ionization energy the students would then have to use the equation E = hcL/λ. Apart from having to convert wavenumber into wavelength (or frequency) there is another problem with the NIST data as students will need to be careful to choose the correct value. This is because the ionization energy for the removal of the second electron is also given (381390.2 cm-1) and rather confusingly they also call this the ground state and give the configuration as 1s22s22p6 1S0.

In fact there is no need for any other sources such as NIST to solve these problems. All the information you require is easily available from the IB data booklet for first exams in 2016.

frown Using the data booklet to obtain the wavelength of convergence limits

For any element E = hcL/λ. Where E is the first ionization energy and λ is the wavelength of the relevant convergence limit. It is usual to solve this for hydrogen either to find the convergence limit or the ionization energy.

Since the ionization energy for hydrogen is given in the data booklet, as are the values for h, c and L it can easily be shown that the convergence limit occurs at 91.2 nm. To follow the instructions given in 12.1 students will normally be given the convergence limit and asked to find the ionization energy.

Since the equation is true for all elements and hcL is a constant it follows that:

IE1(CL)1 = IE2(CL)2

where IE1 and IE2 are the first ionization energies of two different elements and (CL)1 and (CL)2 are their respective relevant convergence limits.

You can therefore easily find the convergence limit for each element simply from its ionization energy which is given in Section 8 of the IB data booklet. Just divide 1312 x 91.2 by the first ionization energy and it will give the convergence limit in nanometres.

For example, the first ionization energy of sodium is 496 kJ mol-1. By doing the sum (1312 x 91.2) / 496 the convergence limit comes out to be 241.4 nm. If you want to check that this is correct, convert it to centimetres and find the reciprocal to give a wavenumber of 4.14 x 104 cm-1 which compares exactly with the NIST value of 4.14 x 104 cm-1. To check it also works for atoms with more than one electron in its outer shell I've randomly done the calculations for carbon (110 nm), fluorine (71.2 nm) and iron (157 nm). In each case the values agree extremely closely with the values given by the NIST source.

What concerns me a little is that all students are being asked to do is manipulate the two equations E = hf and c=λf without really having to show they understand the underlying chemistry. I would ask them which electron is being removed from the sodium atom and therefore which convergence limit is the relevant one. Hopefully they would give the answers 3s1 and the relevant convergence limit is for the series n = 3 to n = 4, n = 5, n = 6 ...... n = infinity. This then begs the Objective 3 question "Suggest why this convergent limit is in the ultraviolet-region of the spectrum when for the hydrogen atom the convergence limit for the n = 3 to n = 4, n = 5, n = 6 ...... n = infinity series is in the infrared region of the spectrum".

There is also a good opportunity to explore some Nature of Science here. Would you mark as correct a student who worked out an enthalpy change and gave the answer in cm-1? The IB Chemistry programme places great emphasis on using the correct terminology and units and yet real scientists, i.e. those people who work at The National Institute of Science and Technology, are quite happy to quote an ionization energy in cm-1 and also not include mol-1. Clearly frequency and wavenumber etc. are all related to energy but they are not an actual recognised unit of energy - at least not as the IB sees it.

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