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.