# Physics307L F09:People/Mondragon/Notebook/071121

## calibration

SJK 21:45, 1 December 2007 (CST)

21:45, 1 December 2007 (CST)
You missed the fact that you were supposed to rotate the prism to correct for this mis-calibration! Your data would be much better in that case. (These methods may be appropriate after you've rotated the prism to the best of your abilities. (Also, did you account for the backlash?)

using mercury to calibrate. center of slit, measuring clockwise

calibration data
color actual wavelength measured wavelength
red 690.75nm 634nm
yellow 579nm 545nm
yellow 577nm 542nm
green 546.1nm 525nm
violet 435.8nm 422nm

so the calibration curve is $\displaystyle{ real\,wavelength=1.204205321\times measured\,value-76.8339591 }$

## the hydrogen spectrum

SJK 21:49, 1 December 2007 (CST)

21:49, 1 December 2007 (CST)
Did you really only take one measurement for each color? (After calibrating correctly (above)), you need to take many "independent" measurements at each color to minimize uncertainty.
hydrogen spectum measurements and rough calculations
color Measured Value Actual Wavelength (nm) Actual Wavelength (m) 1/wavelength n $\displaystyle{ 2^{-2}-n^{-2} }$ Rydberg Constant
red 604 650.5060546 6.50506E-07 1537264.708 3 0.138888889 1.1068306E+07
blue green 466.5 484.927823 4.84928E-07 2062162.558 4 0.1875 1.0998200E+07
violet 421 430.1364809 4.30136E-07 2324843.496 5 0.21 1.1070683E+07
faint violet 400.2 405.0890102 4.05089E-07 2468593.259 6 0.222222222 1.1108670E+07

mean value for Rydberg constant=$\displaystyle{ 1.1061*10^{7}\,\mathrm{m}^{-1} }$ standard deviation $\displaystyle{ 4.6*10^{4}\,\mathrm{m}^{-1} }$

## CALCULATIONS

heres the excel file File:L and T Balmer series.xlsx Section above has numbers from the excel file.

here is a more careful analysis

Linefit says the calibration data fits the line $\displaystyle{ actual\,wavelength=(1.204205 \pm 0.042636)*measured\,wavelength-(76.833959 \pm 22.932003)nm }$

## Error propagation

No data on measurement error was ever taken, so, for any measurement of wavelength the error on the calculated actual value will be

$\displaystyle{ actual\,wavelength\,error=0.042636*measured\,wavelength+22.932003nm }$

The calculated Rydberg constant is related to the measured wavelength value thusly
\displaystyle{ \begin{align} \frac{1}{\lambda} &= R_\mathrm{H}\left(\frac{1}{2^2} - \frac{1}{n^2}\right), n=3,4,5,...\\ R_\mathrm{H} &= \frac{1}{\left(\frac{1}{2^2} - \frac{1}{n^2}\right)\lambda}\\ R_\mathrm{H} &= \frac{1}{\left(\frac{1}{2^2} - \frac{1}{n^2}\right)(m*\lambda_m+b)}\\ \end{align} }
Where $\displaystyle{ \lambda_m }$ is the wavelength measured by the instrument, $\displaystyle{ m }$ is the slope of the calibration curve, and $\displaystyle{ b }$ is the y intercept of the curve. Because of the uncertainty in the fit, the calculated value of the Rydberg constant will have uncertainty.

$\displaystyle{ \Delta R_\mathrm{H} = \frac{1}{\left(\frac{1}{2^2} - \frac{1}{n^2}\right)(m*\lambda_m+b)^2}(\Delta m*\lambda_m+\Delta b)= R_\mathrm{H} \frac{\Delta m*\lambda_m+\Delta b}{m*\lambda_m+b}=R_\mathrm{H} \frac{\Delta \lambda}{\lambda} }$
The uncertiainty in RH is just RH times the relative uncertainty in the calculated wavelength.

hydrogen spectum measurements and rough calculations
color n Accepted Wavelength (nm) Measured Value Calulated Actual Wavelength (nm) Calculated Error(nm) Relative Error Actual Wavelength (m) Rydberg Constant Rydberg uncertainty
red 3 656 604 650.506 48.684 0.074840 6.50506E-07 1.10683E+07 8.284E+05
blue green 4 486 466.5 484.928 44.822 0.088305 4.84928E-07 1.09982E+07 9.712E+05
violet 5 434 421 430.136 40.882 0.095044 4.30136E-07 1.10707E+07 1.0522E+06
faint violet 6 410 400.2 405.089 39.994 0.098731 4.05089E-07 1.11087E+07 1.0968E+06

Average for Rydberg constant= $\displaystyle{ 1.10592*10^7\,\mathrm{m}^{-1} }$. Uncertainty of average = $\displaystyle{ 9.64*10^5\,\mathrm{m}^{-1} }$. Accepted value = $\displaystyle{ 10 967 758.341 \pm 0.001\,\mathrm{m}^{-1} }$