Physics307L F09:People/Mondragon/Notebook/071121
calibration
SJK 21:45, 1 December 2007 (CST)
using mercury to calibrate. center of slit, measuring clockwise
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 [math]\displaystyle{ real\,wavelength=1.204205321\times measured\,value-76.8339591 }[/math]
the hydrogen spectrum
SJK 21:49, 1 December 2007 (CST)
color | Measured Value | Actual Wavelength (nm) | Actual Wavelength (m) | 1/wavelength | n | [math]\displaystyle{ 2^{-2}-n^{-2} }[/math] | 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=[math]\displaystyle{ 1.1061*10^{7}\,\mathrm{m}^{-1} }[/math] standard deviation [math]\displaystyle{ 4.6*10^{4}\,\mathrm{m}^{-1} }[/math]
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 [math]\displaystyle{ actual\,wavelength=(1.204205 \pm 0.042636)*measured\,wavelength-(76.833959 \pm 22.932003)nm }[/math]
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
- [math]\displaystyle{ actual\,wavelength\,error=0.042636*measured\,wavelength+22.932003nm }[/math]
The calculated Rydberg constant is related to the measured wavelength value thusly
[math]\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} }[/math]
Where [math]\displaystyle{ \lambda_m }[/math] is the wavelength measured by the instrument, [math]\displaystyle{ m }[/math] is the slope of the calibration curve, and [math]\displaystyle{ b }[/math] is the y intercept of the curve. Because of the uncertainty in the fit, the calculated value of the Rydberg constant will have uncertainty.
[math]\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} }[/math]
The uncertiainty in RH is just RH times the relative uncertainty in the calculated wavelength.
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= [math]\displaystyle{ 1.10592*10^7\,\mathrm{m}^{-1} }[/math]. Uncertainty of average = [math]\displaystyle{ 9.64*10^5\,\mathrm{m}^{-1} }[/math]. Accepted value = [math]\displaystyle{ 10 967 758.341 \pm 0.001\,\mathrm{m}^{-1} }[/math]