# User:Ginevra Cochran/Notebook/Physics 307L/Balmer series

Steve Koch 22:40, 21 December 2010 (EST):Good primary lab notebook.

## Purpose

The purpose of this lab is to measure the Balmer series of hydrogen and deuterium (as well as the Rydberg constant) using a constant-deviation spectrometer.

## Safety

• Avoid being electrocuted by faulty wiring. Turn off power supply before unplugging it.
• Don't break the glass tubes, especially not the one filled with mercury!
• Be gentle with the equipment.

## Equipment

• constant deviation spectrometer
• spectrum tube power supply Model SP200, 5000 V, 10mA.
• spectrum tubes (mercury, hydrogen, deuterium)

## Setup

We pointed the slit end of the spectrometer at the power source, slotted in a glass tube, and adjusted the slit width and focal length to find a good balance between the number and sharpness of the spectral lines we could see. This setup is outlined in Professor Gold's manual.

## Procedure

We followed the procedure described in Professor Gold's manual. We placed the mercury tube in the power supply and allowed it to heat up for a few minutes, then recorded the dial nanometer reading for spectral lines of mercury given in the manual, starting with red and moving to the right to avoid gear backlash. We did this for every set of measurements. The calibration values for mercury are given in our data table below. Once we had recorded these measurements, we recorded the Balmer spectral lines for our hydrogen and deuterium tubes - 4 lines for hydrogen, 3 for deuterium, and 5 trials each time, always moving from left to right.

## Analysis

I used a best-fit line to determine the calibrating factor for each measured wavelength.

The Rydberg formula as given by Wikipedia is:

• $\displaystyle{ \frac{1}{\lambda }=R(\frac{1}{m^2}-\frac{1}{n^2}) }$
• $\displaystyle{ m=1,2,3,...\,\! }$
• $\displaystyle{ n=2,3,4,5,...\,\! }$

I calculated the accepted value of Rydberg's constant found in Wikipedia.

• $\displaystyle{ R=\frac{\mu e^4}{8\epsilon _0^2ch^3}\, }$
• $\displaystyle{ \mu\, }$ is the reduced mass, which is equal to $\displaystyle{ (m_e*m_p)/(m_e+m_p) }$ for hydrogen and $\displaystyle{ (m_e*(m_p+m_n))/(m_e+m_p+m_n) }$ for deuterium. I calculated the accepted Rydberg constant using the Google document listed above, and found $\displaystyle{ R_H = 1.0902268 *10^7 }$ 1/m and $\displaystyle{ R_D = 1.0905241 *10^7 }$ 1/m.

The transitions for the four visible lines of the Balmer series, according to Wikipedia, are $\displaystyle{ n=6\rightarrow n=2\,\! }$, $\displaystyle{ n=5\rightarrow n=2\,\! }$, $\displaystyle{ n=4\rightarrow n=2\,\! }$, and $\displaystyle{ n=3\rightarrow n=2\,\! }$. In my spreadsheet, I calculated our average wavelength for each of these transitions in hydrogen and deuterium:

• $\displaystyle{ n=6\rightarrow n=2\,\! }$
• $\displaystyle{ \lambda_H = 420.32 nm\,\! }$
• $\displaystyle{ \lambda_D =n/a\,\! }$
• $\displaystyle{ n=5\rightarrow n=2\,\! }$
• $\displaystyle{ \lambda_H = 435.69 nm\,\! }$
• $\displaystyle{ \lambda_D = 435.72 nm\,\! }$
• $\displaystyle{ n=4\rightarrow n=2\,\! }$
• $\displaystyle{ \lambda_H = 484.95 nm\,\! }$
• $\displaystyle{ \lambda_D = 484.81 nm\,\! }$
• $\displaystyle{ n=3\rightarrow n=2\,\! }$
• $\displaystyle{ \lambda_H =653.77 nm\,\! }$
• $\displaystyle{ \lambda_D =651.81 nm\,\! }$

Using the Rydberg formula, I calculated the Rydberg constants for these measured wavelengths.

• $\displaystyle{ n=6\rightarrow n=2\,\! }$
• $\displaystyle{ R_H = 1.0706205*10^7 m^{-1}\,\! }$
• $\displaystyle{ R_D =n/a\,\! }$
• $\displaystyle{ n=5\rightarrow n=2\,\! }$
• $\displaystyle{ R_H = 1.0929663*10^7 m^{-1}\,\! }$
• $\displaystyle{ R_D = 1.0928739*10^7 m^{-1}\,\! }$
• $\displaystyle{ n=4\rightarrow n=2\,\! }$
• $\displaystyle{ R_H = 1.0997799*10^7 m^{-1}\,\! }$
• $\displaystyle{ R_D = 1.1000934*10^7 m^{-1}\,\! }$
• $\displaystyle{ n=3\rightarrow n=2\,\! }$
• $\displaystyle{ R_H =1.1013074*10^7 m^{-1}\,\! }$
• $\displaystyle{ R_D =1.1046079*10^7 m^{-1}\,\! }$

Averaging these values for hydrogen and deuterium, I obtained

• $\displaystyle{ R_Haverage = 1.0911685 +/- 0.0070852*10^7 m^{-1}\,\! }$
• $\displaystyle{ R_Daverage = 1.0991917 +/- 0.0018430*10^7 m^{-1}\,\! }$

## Error

• Our percent errors:
• $\displaystyle{ \% error=\frac{R_{accepted}-R_{measured}}{R_{accepted}} }$
• $\displaystyle{ \% error_H\approx 0.086%\,\! }$
• $\displaystyle{ \% error_D\approx 0.795%\,\! }$

These errors could have resulted from us choosing the wrong spectral line to record or from turning the gear in both directions and causing backlash.

## Sources

Cristhian Carrillo was my lab partner for this lab. I referenced Professor Gold's manual and the Wikipedia articles on the Rydberg formula, the Rydberg constant, and the Balmer series.Peng helped us figure out how to focus the spectrometer correctly. We also received help from Professor Koch and Katie. The photos for this lab are from Alex Andrego's lab notebook.