User:Kirstin Grace Harriger/Notebook/Physics 307L/Balmer Series

Steve Koch 04:02, 21 December 2010 (EST):Very good primary notebook

Concepts



 * Balmer Series: The Balmer Series is of one of a set of six different series describing the spectral line emissions of the hydrogen atom. The Balmer series is calculated using the Balmer formula which was discovered by Johann Balmer in 1885. The visible spectrum of light from hydrogen displays four wavelengths, 410.2 nm, 434.1 nm, 486.1 nm, and 656.3 nm. These correspond to electron transitioning from n = 6,5,4,3 to n = 2 respectively, where n is the principal quantum number of the electron. These are the first four wavelengths in the Balmer Series.

Safety Concerns

 * High voltage on power supply, possibility of shock.
 * Sensitive crystal, do not drop or over tighten clamp.
 * Bulbs are fragile, be careful not to drop.
 * Bulbs get hot, be careful when handling so as not to burn yourself.
 * Spectrometer is very old and precise equipment, do not force or jam things.

Equipment

 * Constant Deviation Spectrometer
 * Spectrum Tube Power Supply Model SP200, 5000 volts, 10MA.
 * Mercury, Hydrogen and Deuterium tubes

Set Up

 * Before turning out the lights or mounting the bulb, focus the cross-hairs by pulling the eyepiece in and out.
 * Mount the mercury bulb and turn it on, being careful not to shock yourself.
 * Open the slit width to a wide opening and place the bulb close to the slit, approximately 1 cm or less away.
 * Change the wavelength positioner (rotates the crystal) until you can see a spectrum line.
 * Look through the eyepiece and turn the knurled nob to focus the slit.
 * Adjust the slit width so that it appears to be a thin line with enough intensity to see it well.

Procedure



 * Calibration
 * Based on the color you see, use the table of wavelengths to find out what wavelength you are looking at.
 * Set the wavelength positioner to that wavelength.
 * Turn the crystal manually so that the line appears in the cross-hairs and coincides with the wavelength positioner value.
 * Find each spectrum line and write down the actual value and the measured value and record the difference so that you have a calibration.


 * Calibration Wavelengths and Colors for Mercury Tube
 * 435.8 nm violet
 * 546.1 nm green
 * 577.0 nm yellow
 * 579.0 nm yellow
 * 690.75 nm red


 * Experiment
 * Now mount Hydrogen and Deuterium and measure all of their spectral lines and record their wavelengths.
 * Look up the principle quantum numbers for Hydrogen (Balmer series image works).
 * Use this data with the Balmer equation to calculate R.

Data
We used the differences in the accepted and measured wavelengths for the spectral lines of mercury to get a correction factor for the measurements of Hydrogen and Deuterium. We averaged the differences between the accepted and measured values to get this correction factor, then we added it to the measured wavelengths of the the Hydrogen and Deuterium lines we were looking for. The correction came out to be -0.67. We could not locate the delta line for deuterium.

Analysis
Using the assumption that the mass of the atomic nucleus is infinite compared to the mass of the electron, the Rydberg constant is given by:


 * $$R_\infty = \frac{m_e e^4}{8 \varepsilon_0^2 h^3 c} = 1.097\;373\;156\;852\;5\;(73) \times 10^7 \ \mathrm{m}^{-1}$$

where me is the rest mass of the electron, e is the elementary charge, ε0 is the permittivity of free space, h is the Planck constant, and c is the speed of light in a vacuum.

The Rydberg constant can also be found with the following equation when the wavelengths and corresponding transitions are known:


 * $$\frac{1}{\lambda }=R(\frac{1}{2^2}-\frac{1}{n^2})$$

where n=3, 4, 5, 6 in the Balmer Series.

Solving for R gives:


 * $$R=\frac{4n^2}{\lambda(n^2-4)}\,\!$$

Calculating R for each wavelength for the first three terms in the Balmer series for hydrogen and deuterium, and then averaging the results gives:


 * $$R_{Hydrogen}= (R_H-alpha + R_H-beta + R_H-gamma)/3 = (10953407.40062 + 11017462.67835 + 11001027.49597)/3 = 10990630.85831 m^{-1}$$


 * $$R_{Deuterium}= (R_D-alpha + R_D-beta + R_D-gamma)/3 = (10970091.26506 + 10989086.46351 + 11014513.82487)/3 = 10991230.51781 m^{-1}$$

The known values for $$R_{Hydrogen}$$ and $$R_{Deuterium}$$ are found by using the following equation:


 * $$R_M = \frac{R_\infty}{1+m_e/M},$$

where me is the rest mass of the electron, and M is the mass of the atomic nucleus.

These values are:


 * $$R_{Hydrogen}= 10967758.3406 m^{-1}\ \ $$


 * $$R_{Deuterium}=10970746.1986 m^{-1} \ \ $$

The calculated percent errors are given by:
 * $$\% error=\frac{R_{accepted}-R_{measured}}{R_{accepted}}$$

These values are:


 * $${Hydrogen}\approx0.21%\,\!$$


 * $${Deuterium}\approx0.19%\,\!$$

Resources

 * 1) Prof. Gold's Lab Manual
 * 2) Wikipedia Article on the Balmer Series
 * 3) Wikipedia Article on the Rydberg Constant
 * 4) Wikipedia Article on the Rydberg Formula

Collaboration

 * 1) My lab partner Brian Josey
 * 2) Emran Qassem who let me use his lab pictures