**THE SPECTROMETER**

The applet below illustrates the functioning of a spectrometer in a very schematic way. For the purpose of this animation, the source will produce monochromatic light in the visible region of the spectrum. The beam passes through a cell containing a sample and is detected by an appropriate device coupled to a chart-recorder. (Of course, modern instruments would display the results on a computer screen!)

The "Light" button turns out the "room" lights to make it easier to see the beam. You can start, pause, and stop the scan using the buttons so marked. Once started, the scans will repeat automatically if you do not interrupt them.

Observe the beam as it passes through the sample and how this is recorded on the chart. Then answer the questions which follow.

What property of the light varies as it passes through the sample; the wavelength or the intensity?

**Question 2.**

In your report, re-sketch the graph which is produced by the spectrometerwith the axes labeled appropriately.

A spectrometer entirely analogous to the one depicted above can be
constructed to record the microwave absorption spectrum of a gas. In such a case
the source is called a klystron and the radiation, invisible of course, passes
through a metal tube called a wave guide, through the gaseous sample under
study, and on through another wave guide to the detector. A typical microwave
spectrum for isotopically pure ^{1}H^{35}Cl gas is shown below.

**
**

**Question 3.**

Sketch a copy this spectrum and report, as accurately as you can, the position of each of the peaks.

__CALCULATION OF THE BOND LENGTH IN HCl__

The object of this section is to determine the bond length in the molecule HCl.

**Question 4.**

Calculate and report the reduced mass of^{1}H^{35}Cl. Assume a mass of 1.0 Daltons for H and 35.0 Daltons for Cl, and give your answer in kg. (Avagadro's number is 6.02x10^{23}.)

After transformation from the "real" situation (which was shown in Figure 1) to the mathematical model (which was shown in Figure 2), the reduced H/Cl mass rotates around a circle whose radius is equal to the bond length. Unlike the electron in its orbits whose radii differ, the bond length in HCl is treated invariant for the purposes of this "experiment".

**Question 5.**

This is actually not quite correct - can you suggest why?

We have seen that various associated wavelengths are possible. The de Broglie
postulate says that the wavelength is related to the **momentum** of the
particle by the equation:

where l is the wavelength, M.v is the reduced mass
times the velocity, or momentum, and h is Planck's constant,
6.63x10^{-34} J s. The lines in the microwave spectrum that the computer
shows you correspond with the wavelength of the photons absorbed when the
reduced mass decreases *its* associated wavelength. The important quantity
here is the **energy** of the photons absorbed, which is equal to the change
in kinetic energy of the reduced mass. This is given by the equation:

**Question 6.**

Show that the kinetic energy can be expressed in terms of the wavelength by the equation:

K = h ^{2}/(2.M.l^{2}) ...............(4)

**Question 7.**

Use the equation you derived in Dry lab 1, Question 10 to eliminate l, leaving a final equation in j, r, M and numerical constants. Collect all the numerical constants and the two properties M and r of the H-Cl molecule in a constant B_{HCl}. Report its value.

You should now have an equation of the form:

**Question 8.**

Show that your B_{HCl}(called B from now on) has units of Joules (kg m^{2}s^{-2}). (If it does not, do not proceed until you have corrected it!)

We can now return to the data that you recorded for the microwave spectrum of
HCl. It is the line spacing which is important to us. We can understand how
these lines arise given one further piece of information: changes in j are
governed by the *selection rule* that Dj = +1 for
an absorption (or -1 for emission). **Other transitions are not allowed.**
The energy of the lines in the microwave spectrum correspond to differences
between rotational energy levels whose quantum numbers differ by 1. The next
"question" develops this idea.

**Question 9.**

- Sketch a 1-dimensional energy level diagram like the one shown in figure 4 (below) labeling the energy axis in units of B. The first two levels, for j = 0 and 1, are drawn in (in red) as an example.
- Mark each
allowedtransition. The first (j = 0 to j = 1) is shown (in green) for you.- Each transition will give rise to a spectral line like the ones in Figure 3. In units of B, what is the spacing
betweenthe spectral lines?

**Question 10.**

Now use your measured peak positions (from question 4) to calculate the value of B as precisely as you can - think about what would be the best strategy. Bear in mind that there is no guarantee that you are seeing all the lines, in particular, the lowest energy line recorded might not be the line corresponding to the transition j = 0 to j = 1.To convert from cm

^{-1}to Joules use this equation:

E(J) = h.c.n ................(6) where c is the speed of light (3x10

^{8}m s^{-1}), and n is the energy in units of cm^{-1}(called wave numbers).

**Question 11.**

Finally, calculate the value of the bond length from the value you have obtained for B.

**POSTSCRIPTS**

- Modern quantum mechanics would lead to one revision of the above
mathematics, which is necessary to make it consistent with the Heisenberg
principle. In place of j
^{2}, modern spectroscopy books would give j(j+1). This makes no difference to the results. Go back to Figure 4 and recalculate the line spacings in terms of B to convince yourself of this. This change prevents the molecule from ever being attributed zero rotational energy. - This method is applicable to all small molecules: the expression for B changes for molecules other than diatomics since more parameters are needed to describe them, and for molecules which are not linear, up to 2 additional quantum numbers are required. The most accurate bond length and angle data that are available to us come from microwave studies.