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 = 11.64 µm-1For nu= 4, R = 10.35 µm-1For nu= 5, R = 10.56 µm-1For nu= 6, R = 11.87 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 = 11.64 µm-1For nu= 4, R = 10.35 µm-1For nu= 5, R = 10.56 µm-1For nu= 6, R = 11.87 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 the mean error.
Step 1: Calculate the average R
The average R is the sum of all R values divided by the number of R values.
R_avg = (R1 + R2 + R3 + R4) / 4
R_avg = (11.64 µm-1 + 10.35 µm-1 + 10.56 µm-1 + 11.87 µm-1) / 4
R_avg = 44.42 µm-1 / 4
R_avg = 11.105 µm-1
Step 2: Calculate the mean error
The mean error is the sum of the absolute differences between each R value and the average R, divided by the number of R values.
Mean error = (|R1 - R_avg| + |R2 - R_avg| + |R3 - R_avg| + |R4 - R_avg|) / 4
Mean error = (|11.64 µm-1 - 11.105 µm-1| + |10.35 µm-1 - 11.105 µm-1| + |10.56 µm-1 - 11.105 µm-1| + |11.87 µm-1 - 11.105 µm-1|) / 4
Mean error = (0.535 µm-1 + 0.755 µm-1 + 0.545 µm-1 + 0.765 µm-1) / 4
Mean error = 2.6 µm-1 / 4
Mean error = 0.65 µm-1
So, the uncertainty in the average R is approximately 0.65 µm-1.
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