# User:David K. O'Hara/Notebook/physics 307 lab/balmer series

Lab Balmer Series

SJK 21:36, 14 November 2009 (EST)

21:36, 14 November 2009 (EST)
Excellent primary lab notebook! It is very easy to follow what you're doing and your analysis.

## Theory

Lab Objectives

In this lab we will calibrate an optical spectrometer using the known mercury spectrum.
Study the Balmer Series in Hydrogen spectrum.
Determine the Rydberg Constant for Hydrogen.
Last, compare Hydrogen with Dueterium.

Using a constant deviation spectrometer, which has the characteristic that the angle of incidence and the angle of exit are the same for all wavelengths when the prism is properly oriented. All that is required for the spectral analysis of light is to rotate the prism relative to the incident light keeping the incident ray and the axis of the analyzing telescope fixed at 90 degrees. The rotation of the prism is calibrated with a known source (mercury in this case), and an interpolation of the known lines is used for the final calibration. See this link about constant deviation prisms [Pellin-Broca Prisms].

## Equipment

• Adam Hilger London Spectrometer. Serial Number 12610
• Spectrum Tube Power Supply Model SP200 5000V 10 mAmps electro technic products
• Spectrum Tube, Mercury Vapor S-68755-30-K
• Spectrum Tube, Hydrogen S-68755-30-G
• Spectrum Tube, Deuteruim S-68755-30-E
• Spectrum Tube, Krypton S-68755-30-?

## Safety

Cautions for electric shock due to the high voltage (5000v)the power supply runs across the sample tubes to make them light up.
Spectrometer is very old and not built to stand up to rough handling.
Sample tubes are glass and need careful handling to prevent breaking.
One of the samples is mercury which is toxic if released out of the tube and would be quite hazardous to anyone in the vicinity. Any lack of containment would require containment and proper clean-up/disposal immediately.
Lastly, the sample tubes do get quite hot as they are lit for periods of time. You have to let them cool before you can safely change from one tube to another.

## Procedure

Using the spectrometer shown on the right:

First step is to calibrate the spectrometer using the mercury lamp.

1. Power up the mercury tube and let it warm up for about 5 minutes
2. With the spectrometer slit wide(.5-1mm) find a line of the Hg spectrum.
3. Narrow the slit until the mercury line becomes sharp and narrow.
4. Locate all mercury lines possible.

wavelength (nm) color measured λ measured λ (2nd run) measured λ (3rd run)
404.7 deep violet 404.7 404.9 not visible
435.8 violet 434.2 435.5 435.7 436.2
skip blue-green 488.8 490.9 491.1 491
546.1 green 541.1 545 547 546.2
577.0 yellow 574.5 578 578.5 577.8
579.0 yellow na na 581.3 579.5
690.75 red 695 difficult to see not visible not visible

First run on calibrating the prism, I left the slit wide to allow plenty of light through and measured from the center of each line. This particularly affected the reading on the yellow lines as the resolution was too poor to pick out both lines. For the second run I will reduce the slit width and take the readings from the leading edge of the observed lines.
Still had the slit too wide for the second run, yellow lines were still not defined as a duo and noticeable error propagation as it got to the end of the scale. Did seem better lined up than the first set though. But I should have realized that if I could pick a leading edge of the lines, then the lines were too fat. Hopefully the third time is the charm.

For the third run, I have made sure I have the duo of yellow lines, I set the measurement for 546.1nm and then I centered the green line on the crosshairs. Data were taken from 400nm up.

The Fourth run I left everything the same but worked from 700nm down. Made sure to use the same eye (left) for each measurement and tried to reduce all sources of parallax that I could(viewing angle, head pitch, etc.). Seemed to be the truest set of data I took for the mercury lamp.

## data

Getting the Hydrogen lamp, I decided to pass on the one with "blown" written on a piece of paper taped to sample and went with another Hydrogen tube. Again, measurements were taken from 700 nm down.

Take hydrogen Data, should see 4-5 lines, record raw and corrected data.

color Raw measured λ run1 Raw measured λ run2 λ correction
red 657.2 657
blue-green 485.8 486
blue-violet 434 433.7
deep violet 410 410.1

Accepted wavelengths of these lines pulled from this wikipedia page for [balmer series]
Red 656.3 nm
Blue-Green 486.1 nm
Blue 434.1 nm
Violet 410.2 nm

## day 2 measurements

to begin day 2 I started again with calibrating the spectrometer using the mercury lamp. I took the measurement for the green line multiple times to find an error range for that measurement, then I applied that error range to the rest of the lines to make sure all the data I would take would be within that range.

SJK 20:40, 14 November 2009 (EST)

20:40, 14 November 2009 (EST)
If I remember, I did not spend enough time talking with you during the lab, especially to discuss the uncertainty. It's great that you did this for the one line to estimate uncertainty. However, a couple important points: (1) I think you used the standard deviation of the measurements, as opposed to the standard error of the mean (differ by a factor of 1/sqrt(5) ). Second, you need to do this for EACH color! It is quite likely that the SEM depends on the value of the wavelength (as opposed to being constant as you've assumed)
λ expected(nm) 546.1 line(nm) 546.1 line(nm) 546.1 line(nm) 546.1 line(nm) error range (nm) λ measured (nm)
579.0 (yellow) 579.2
577.0 (yellow) 576.8
546.1 (green) 546.0 546.3 546.5 545.8 =+/-.25 545.9
435.8 (violet) 435.6

all the data for the mercury lines is within my error expectation predicted off of the green line measurements.

2nd day hydrogen data set 1 (run 3)

λ measured (nm) w/error margin
656.1 +/-0.25 (red)
485.9 +/-0.25 (cyan)
434.2 +/-0.25 (blue-violet)
410.2 +/-0.25 (faint violet)

2nd day Hydrogen data set 2(4th run)

λ measured (nm) w/error margin
656.1 +/-0.25 (red)
486.0 +/-0.25 (cyan)
434.3 +/-0.25 (blue-violet)
410.3 +/-0.25 (faint violet)

## Dueterium Data

λ measured (nm) w/error margin
656.0 +/-0.25 (red)
485.3 +/-0.25 (cyan)
433.9 +/-0.25 (blue-violet)
409.2 +/-0.25 (faint violet)

Dueterium 2nd Run

λ measured (nm) w/error margin
655.9 +/-0.25 (red)
485.3 +/-0.25 (cyan)
434.1 +/-0.25 (blue-violet)
409.6 +/-0.25 (faint violet)

Dueterium 3rd run

λ measured (nm) w/error margin
655.8 +/-0.25 (red)
485.4 +/-0.25 (cyan)
433.9 +/-0.25 (blue-violet)
409.7 +/-0.25 (faint violet)

Dueterium 4th run

λ measured (nm) w/error margin
656.0 +/-0.25 (red)
485.5 +/-0.25 (cyan)
433.8 +/-0.25 (blue-violet)
409.6 +/-0.25 (faint violet)

## Krypton

The Lab manual also suggests reading the spectrum from a Sodium Lamp which was not available so I substituted Krypton instead of Na. krypton first run

λ measured (nm) w/error margin
586.9 +/-0.25 (orange)
556.5 +/-0.25 (green)
431.5 +/-0.25 (violet)
426.8 +/-0.25 (violet)

krypton second run

λ measured (nm) w/error margin
586.8 +/-0.25 (orange)
556.3 +/-0.25 (green)
431.3 +/-0.25 (blue-violet)
426.8 +/-0.25 (faint violet)

Since the point of the exercise was to see if you could resolve the two sodium lines which are very close together, this part of the lab while interesting to see the characteristic orange and green krypton lines, was quite pointless.

## Analysis

The value for the Rydberg constant is given by the equation:

$\frac{1}{\lambda }=R_{H}(\frac{1}{2^{2}}-\frac{1}{n^{2}})$

With the accepted value for that constant as Rh=1.097272 x 107m-1 [|Ryderg constant wiki page]

To find the value of n for the wavelengths measured I again used this page as a resource [|balmer equation]

This spreadsheet shows values for the Hydrogen and Deuterium lines average wavelength and the standard error of the mean for the measurements File:Balmer data.xlsx

using this equation to find RH $\frac{\frac{1}{\lambda }}{(\frac{1}{2^2}-\frac{1}{n^2})}=R_{H}$

• $\frac{(\frac{1}{656.6nm})}{(\frac{1}{2^2}-\frac{1}{3^2)}}$ = $R_{H}$ for the red spectral line = 1.0965580$*10^{7}m^{-1}$
• $\frac{(\frac{1}{485.925nm})}{(\frac{1}{2^2}-\frac{1}{4^2)}}$ = $R_{H}$ for the cyan spectral line = 1.0953089$*10^{7}m^{-1}$
• $\frac{(\frac{1}{434.05nm})}{(\frac{1}{2^2}-\frac{1}{5^2)}}$ = $R_{H}$ for the blue-violet spectral line = 1.0970866$*10^{7}m^{-1}$
• $\frac{(\frac{1}{410.15})}{(\frac{1}{2^2}-\frac{1}{6^2)}}$ = $R_{H}$ for the violet spectral line = 1.0971595$*10^{7}m^{-1}$

The Rydberg constant that i calculated for hydrogen from my data was 1.0965283e7 m-1 +/-4300

Dueterium data

• $\frac{(\frac{1}{655.925nm})}{(\frac{1}{2^2}-\frac{1}{3^2)}}$ = $R_{D}$ for the red spectral line = 1.0976864$*10^{7}m^{-1}$
• $\frac{(\frac{1}{485.375nm})}{(\frac{1}{2^2}-\frac{1}{4^2)}}$ = $R_{D}$ for the cyan spectral line = 1.0988067$*10^{7}m^{-1}$
• $\frac{(\frac{1}{433.925nm})}{(\frac{1}{2^2}-\frac{1}{5^2)}}$ = $R_{D}$ for the blue-violet spectral line = 1.0974027$*10^{7}m^{-1}$
• $\frac{(\frac{1}{409.525})}{(\frac{1}{2^2}-\frac{1}{6^2)}}$ = $R_{D}$ for the violet spectral line = 1.0966340$*10^{7}m^{-1}$

The Rydberg constant that i calculated using my dueterium data was 1.0976325e7 m-1 +/-4500, slightly different from hydrogen due to the heavier atom.

• Equation formatting taken from David Weiss openwetware page "balmer lab notes"

## Conclusions

The value of the Rydberg constant I calculated for hydrogen was 1.0965283e7 m-1 +/-4300. This compared to the accepted value of the Rydberg Constant of 1.097272 x 107 m-1.

SJK 21:35, 14 November 2009 (EST)

21:35, 14 November 2009 (EST)
As mentioned on your summary page, this is indeed very good accuracy, but you need to statistically compare with the accepted value to say whether it's consistent or not. Also, I think you used R_infinity as opposed to R_hydrogen or R_deuterium

This gives a percent error from the accepted value of .000678 which is the closest I have come to an accepted value in this class.

Possible sources of error: The scale on the spectrometer is not fine enough to be accurate to a tenth of a nanometer so all measurements are off by as much as a quarter nm. The way you set up to view the spectral lines in the scope has a susceptibility to induce a large amount of parallax, especially dealing with such fine measurements. As stated in the lab manual, the tradeoff between resolution vs brightness puts the experimenter in the position of trying to find the optimal condition for viewing the spectral lines.