# Physics307L F09:People/Gibson/Notebook/071105

## Objectives

In this experiment we will observe the Balmer Series of Hydrogen and Deuterium.

• Review basic atomic physics.
• Calibrate an optical spectrometer using the known mercury spectrum.
• Study the Balmer Series in the hydrogen spectrum.
• Determine the Rydberg constant for hydrogen.
• Compare hydrogen with deuterium

## Set Up

We first plugged the mercury tube into the lamp and let it warm up for at least 5 minutes. After this we checked to see if the following values of lambda would result in giving the correct color spectra of mercury i.e. this:

• 404.7 nm (deep violet very hard to see!!)
• 435.8 nm violet
• skip (very weak blue-green)
• 546.1 nm green
• 577.0 nm yellow
• 579.0 nm yellow
• 690.75 nm red

And this is the calibration step. The values of lambda (in nm) are reached by rotation of a knob with specific values of lambda on it.

Our values of lambda for the mercury are:

  Violet:   436.4 nm
Green:    548.1 nm
Yellow 1: 580.2 nm
Yellow 2: 582.1 nm
Red:      Cannot see


## Data

• Hydrogen
Color Violet 1 nm Violet 2 nm Blue nm Red nm
Quantum Number $\displaystyle n=6$ ${\displaystyle n=5}$ ${\displaystyle n=4}$ ${\displaystyle n=3}$
Measurement
1 410.5 432.9 486.5 660.9
2 410.0 433.2 486.9 661.1
3 409.5 433.6 487.2 660.9
4 410.5 433.5 487.8 660.7
5 410.5 433.6 486.5 660.5
6 410.25 433.2 486.5 660.5
7 410.0 433.6 486.5 660.7

• Deuterium
Color Violet 1 nm Violet 2 nm Blue nm Red nm
Quantum Number ${\displaystyle n=6}$ ${\displaystyle n=5}$ ${\displaystyle n=4}$ ${\displaystyle n=3}$
Measurement
1 410.0 432.1 486.0 660.0
2 410.0 433.1 486.1 660.1
3 409.75 433.2 486.5 660.1
4 409.75 433.1 486.5 660.0
5 409.75 433.1 486.2 660.0
6 410.0 433.1 486.2 660.0
7 410.0 433.1 486.6 660.0
• Neon
Color Yellow nm Red nm
Quantum Number ${\displaystyle n=4}$ ${\displaystyle n=3}$
Measurement
1 580.3 642.0
2 585.8 642.0
3 585.6 642.1

## Analysis

${\displaystyle R={\frac {1}{{\frac {\lambda }{4}}-{\frac {\lambda }{n^{2}}}}}}$

• Using excell to find the mean wavelength for each color of Hydrogen and Deuterium, we find:
Color Violet 1 nm Violet 2 nm Yellow nm Red nm
Quantum Number ${\displaystyle n=6}$ ${\displaystyle n=5}$ ${\displaystyle n=4}$ ${\displaystyle n=3}$
Hydrogen
Mean 410.1786 433.3714 486.8429 660.7571
STD Deviation .3740 .2752 .5028 .2225
Deuterium
Mean 409.89 433.12 486.40 660.02
STD Deviation .1336 .3860 .2309 .0487
• Using the formula given above we find that the Ryberg Constant for each color of Hydrogen and Deuterium is:
Color Violet 1 nm Violet 2 nm Yellow nm Red nm
Hydrogen
Rydberg R (m^-1) ${\displaystyle 1.0971\times 10^{7}}$ ${\displaystyle 1.0988\times 10^{7}}$ ${\displaystyle 1.0955\times 10^{7}}$ ${\displaystyle 1.0897\times 10^{7}}$
Deuterium
Rydberg R (m^-1) ${\displaystyle 1.0978\times 10^{7}}$ ${\displaystyle 1.0994\times 10^{7}}$ ${\displaystyle 1.0965\times 10^{7}}$ ${\displaystyle 1.0909\times 10^{7}}$
• Now taking the mean of the measured Rydberg Constant R for each element, we find:
• For Hydrogen: ${\displaystyle R_{H}=\left(1.0953\pm .0039\right)\times 10^{7}{m^{-}1}}$
• For Deuterium: ${\displaystyle R_{H}=\left(1.0962\pm .0037\right)\times 10^{7}{m^{-}1}}$

## Results

The value we found for Hydrogen was:

$\displaystyle R_H = 10 967 758.341 \pm 0.001\,\mathrm{m}^{-1} \$

As one can see, our value does not stray too far from the accepted value:

• For Hydrogen: $\displaystyle R_H=\left(1.0953 \pm .0039\right)\times 10^7 {m^-1}$

The accepted value of the Ryderbeg constant for Deuterium is: $\displaystyle R_D=\left(1.09707 \pm .7\right)\times 10^7 {m^-1}$

Again, our value was very close to the accepted value:

• For Deuterium: $\displaystyle R_D=\left(1.0953 \pm .0039\right)\times 10^7 {m^-1}$

## Analysis

From the way we performed our experiment, the results could not have been better. It was very difficult trying to see the lines for each, but by keeping our precision high when recording our values proved to help us in the end. Taking the data could involve some sort of error seeing as aligning the lines on the cross hairs exactly was quite eye straining.

All our values we're very close to the accepted ones we found online so it was comforting to say the least that this equipment (aged) could still give good data given the situation.