Posted: July 12th, 2022

Conceptual Physics II

Introduction
In this lab, you will investigate how a grating spectrometer works to measure the visible spectrum of the hydrogen atom, along with some other elements. As we discussed in lecture, a spectrometer is an instrument used for measuring the wavelength (or frequency) of electromagnetic radiation. It consists of several elements, the main ones being a slit, a dispersive element like a prism, and a detector of the radiation. The slit clearly defines the source of the radiation, the dispersive element divides the radiation into its component frequencies/wavelengths based on their angle of refraction, and finally the detector detects the radiation.
One of the spectra that you will study today is the emission spectrum emitted by the hydrogen atom. This particular spectrum is called the Balmer Series. The Balmer series is the name given to a series of spectral emission lines of the hydrogen atom that result from electron transitions from higher levels down to the energy level with principal quantum number 2, like shown in the diagram below:

The Balmer Series is given by the Rydberg formula. This formula can be used to calculate the reciprocals of the wavelengths in the series. Specifically:
Formula 1
Here, R is called the Rydberg constant and has a value , while n takes on integer values beginning with 3. There are typically three or four lines visible in the Balmer Series, corresponding to the three or four transitions that excited electrons can make as they move from a higher to a lower energy state in the Hydrogen atom. n =1 corresponds to the lowest energy state available in the Hydrogen atom, and each subsequent value of n corresponds to the next available energy level in the atom.
When an electron in the Hydrogen atom has been excited to the third energy level (n =3), it can make a transition to the lowest energy level, emitting a photon with frequency in the red region. This shows up as a red line in the emission spectrum of Hydrogen and is known as the Hydrogen-alpha line. The line corresponding to is green, and the last two lines visible, corresponding to and , respectively, are violet. Lines for values of n beyond 6 are in the ultraviolet and are not visible.
In this lab, you will first watch a video showing the emission spectra of a variety of elements and then use a simulator to measure the Balmer spectrum of Hydrogen. You will then compare your measured values to those computed from the Rydberg formula.

Procedure

Part I

1. Watch the following short video, which shows the emission spectra of different elements, as well as a thermal emitter, as light from each is passed through a diffraction grating: https://www.youtube.com/watch?v=2ZlhRChr_Bw

2. Now answer the following questions, based on the video you just watched:

• (4 pts) How many spectral lines show up in the emission spectrum of Hydrogen? What colors are they?

• (2 pts) Does the emission spectrum of Helium consist of fewer or more spectral lines than Hydrogen?

• (2 pts) How many spectral lines show up in the emission spectrum of Mercury? Are they the same or different than the lines in the Hydrogen spectrum?

• (4 pts) What do you notice about the emission spectrum of Carbon Dioxide? (Hint: Carbon Dioxide is a molecule made up of two elements; Carbon, and Oxygen. Would you expect to see more or less spectral lines in the spectrum of a molecule, as compared to those from a single element like Carbon or Oxygen?)

• (4 pts) Finally, what do you notice about the emission spectrum of a thermal emitter like an incandescent lamp? Why is this spectrum different than that of a single element?

Part II

Now that you have a better understanding of what the emission spectra of different elements look like, let’s use a simulator to analyze the absorption spectrum of Hydrogen in a bit more detail.

1. Go to https://physics.bu.edu/~duffy/HTML5/emission_spectra.html and pick “Hydrogen”, just below the Emission Spectra image. You should see 4 spectral lines show up. These correspond to the four possible transitions (that fall in the visible part of the spectrum) an electron can make in the Hydrogen atom.

2. (4 pts) For each of the four spectral lines in the Hydrogen spectrum, record the corresponding wavelength in the “Measured” column in the table below.

3. (12 pts) Calculate the wavelength of the first four lines in the Balmer Series using the Rydberg formula (Equation 1) and record your results in the “From Formula 1” column in the table. Don’t forget to take the reciprocal of the wavelength as the last step in your calculations. Additionally, remember to convert your answer from m to nm so that it matches your measured values.
For reference, 1 nm = 1 x 10-9 m.
Show all calculations on a separate sheet of paper.

• (8 pts) Calculate the percent difference between each of your measurements and the value calculated from the Rydberg formula. (The percent difference shows the deviation of the measured value from the formula value and can be calculated by equation:

Formula Wavelength – Measured Wavelength
_______________________________________ x 100 = % Difference

Formula Wavelength

Color n Wavelength (nm)
Measured Wavelength (nm)
From Formula 1 % Difference
Red
3
Green
4
Violet 1
5
Violet 2
6

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