In the spectrum of a specific element, there is a line with a wavelength of 656 nm. Use the Rydberg equation to calculate the value of n for the higher energy Bohr orbit involved in the emission of this light. Assume the value for the lower energy orbit equals 2.
1/wavelength = R(1/4 - 1/n^2)
1/4 is 1/2^2
To solve this problem, we can use the Rydberg equation, which relates the wavelength of light emitted or absorbed by an electron transition in a hydrogen-like atom to the difference in energy levels. The Rydberg equation is given by:
1/λ = R * (1/n₁² - 1/n₂²)
Where λ is the wavelength of light, R is the Rydberg constant (1.0973731568508 x 10^7 m⁻¹), n₁ is the lower energy level, and n₂ is the higher energy level.
In this case, the lower energy orbit (n₁) is given as 2. We need to calculate the value of n₂ for the higher energy Bohr orbit involved in the emission of this light.
Let's substitute the given values into the Rydberg equation and solve for n₂:
1/λ = R * (1/n₁² - 1/n₂²)
Since the wavelength (λ) is given as 656 nm, we need to convert it to meters:
λ = 656 nm = 656 x 10⁻⁹ m
Now, we can substitute the values into the equation:
1/(656 x 10⁻⁹ m) = (1.0973731568508 x 10^7 m⁻¹) * (1/2² - 1/n₂²)
Simplifying the equation:
1/(656 x 10⁻⁹ m) = (1.0973731568508 x 10^7 m⁻¹) * (1/4 - 1/n₂²)
Solving for n₂:
1/n₂² = 1/4 - 1/(1.0973731568508 x 10^7 m⁻¹) * (656 x 10⁻⁹ m)
Now, isolate n₂ by taking the reciprocal of both sides:
n₂² = 4 / (1/4 - 1/(1.0973731568508 x 10^7 m⁻¹) * (656 x 10⁻⁹ m))
Finally, take the square root of both sides to get the value of n₂:
n₂ = √[4 / (1/4 - 1/(1.0973731568508 x 10^7 m⁻¹) * (656 x 10⁻⁹ m))]
Evaluating this expression will give you the value of n₂ for the higher energy Bohr orbit involved in the emission of the given light.