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).In this series, the transition from nu= 3 to nl= 2 produces the photon with the lowest energy, this corresponds to the line with the longest wavelength. This is measured to be λ = 658.1 nm.What value of the Rydberg constant R is obtained using these measurements?
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).In this series, the transition from nu= 3 to nl= 2 produces the photon with the lowest energy, this corresponds to the line with the longest wavelength. This is measured to be λ = 658.1 nm.What value of the Rydberg constant R is obtained using these measurements?
Solution
The Rydberg formula for the Balmer series is given by:
1/λ = R * (1/nl^2 - 1/nu^2)
where: λ is the wavelength of the light, R is the Rydberg constant, nl is the lower energy level, and nu is the upper energy level.
Given that λ = 658.1 nm = 658.1 * 10^-9 m (since 1 nm = 10^-9 m), nl = 2, and nu = 3, we can substitute these values into the Rydberg formula and solve for R:
1/(658.1 * 10^-9 m) = R * (1/2^2 - 1/3^2)
R = 1/(658.1 * 10^-9 m * (1/4 - 1/9))
R = 1/(658.1 * 10^-9 m * 5/36)
R = 36/(658.1 * 10^-9 m * 5)
R = 7.2/(658.1 * 10^-9 m)
R = 1.095 * 10^7 m^-1
So, the Rydberg constant R obtained using these measurements is approximately 1.095 * 10^7 m^-1.
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