The spectrum from a hydrogen vapour lamp is measured and four lines in the visible light range are observed. These lines are the so-called Balmer series, where an electron makes a transition from a higher level to the second energy level (nl= 2).From the measurements of the wavelengths of these four lines, four determinations of the Rydberg constant R were made. These are:For nu= 3, R = 9.96 µm-1For nu= 4, R = 9.65 µm-1For nu= 5, R = 9.72 µm-1For nu= 6, R = 11.9 µm-1An average value of R can be found. Calculate the uncertainty in this average R (use the mean error technique).Express you answer in µm-1 to two decimal places.
Question
The spectrum from a hydrogen vapour lamp is measured and four lines in the visible light range are observed. These lines are the so-called Balmer series, where an electron makes a transition from a higher level to the second energy level (nl= 2).From the measurements of the wavelengths of these four lines, four determinations of the Rydberg constant R were made. These are:For nu= 3, R = 9.96 µm-1For nu= 4, R = 9.65 µm-1For nu= 5, R = 9.72 µm-1For nu= 6, R = 11.9 µm-1An average value of R can be found. Calculate the uncertainty in this average R (use the mean error technique).Express you answer in µm-1 to two decimal places.
Solution
To calculate the uncertainty in the average R, we first need to calculate the average R and then find the mean error.
Step 1: Calculate the average R
Add up all the values of R and divide by the number of values:
R_avg = (9.96 + 9.65 + 9.72 + 11.9) / 4 R_avg = 10.5575 µm^-1
Step 2: Calculate the mean error
The mean error is the average of the absolute differences between each value and the average value. So, we calculate the absolute differences:
|9.96 - 10.5575| = 0.5975 |9.65 - 10.5575| = 0.9075 |9.72 - 10.5575| = 0.8375 |11.9 - 10.5575| = 1.3425
Then we calculate the average of these differences:
Mean error = (0.5975 + 0.9075 + 0.8375 + 1.3425) / 4 Mean error = 0.92125 µm^-1
Rounding to two decimal places, the uncertainty in the average R is approximately 0.92 µm^-1.
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