- The shortest wavelength (λshortest) can be calculated using the Rydberg formula with n1 = 8 and n2 = 2.
- The longest wavelength (λlongest) can be calculated using the Rydberg formula with n1 = 8 and n2 approaching infinity.
When a hydrogen atom is in its eighth excited state, the shortest and longest wavelengths (in meters) of the photons it can emit can be determined using the Rydberg formula and the Balmer series.
The Rydberg formula is given as:
1/λ = R * (1/n1^2 - 1/n2^2)
Where λ is the wavelength, R is the Rydberg constant (approximately 1.097 x 10^7 m^-1), and n1 and n2 are the principal quantum numbers.
In the Balmer series, the transitions occur from higher energy levels to the second energy level (n2 = 2).
To find the shortest wavelength, we consider the transition from the eighth excited state (n1 = 8) to the second energy level (n2 = 2). Plugging these values into the Rydberg formula, we get:
1/λshortest = R * (1/2^2 - 1/8^2)
To find the longest wavelength, we consider the transition from the eighth excited state (n1 = 8) to the second energy level (n2 = 2). Plugging these values into the Rydberg formula, we get:
1/λlongest = R * (1/2^2 - 1/∞^2)
As n2 approaches infinity, the term 1/n2^2 approaches zero, resulting in the longest wavelength.