use the rydberg equation to calculate the wavelength in A of the photon absorbed when a hydrogen atom undergoes a transition from n=7 to n=9
To calculate the wavelength of a photon absorbed in an electronic transition of a hydrogen atom using the Rydberg equation, follow these steps:
Step 1: Review the Rydberg equation.
The Rydberg equation is given by:
1/λ = R * (1/n₁² - 1/n₂²)
where:
- λ is the wavelength of the absorbed or emitted photon
- R is the Rydberg constant (1.0974 x 10^7 m⁻¹)
- n₁ is the initial energy level or principal quantum number
- n₂ is the final energy level or principal quantum number
Step 2: Identify the initial and final energy levels (n₁ and n₂).
In this case, the transition is from n = 7 (initial) to n = 9 (final).
Step 3: Substitute the values into the Rydberg equation.
1/λ = R * (1/7² - 1/9²)
Step 4: Simplify and solve for λ.
1/λ = R * (1/49 - 1/81)
1/λ = R * (32/4900 - 20/4900)
1/λ = R * (12/4900)
1/λ = 0.0024489 R
Now, take the reciprocal of both sides of the equation:
λ = 1/(0.0024489 R)
Step 5: Calculate the wavelength (λ).
Now, substitute the value of the Rydberg constant (R = 1.0974 x 10^7 m⁻¹) into the equation:
λ = 1/(0.0024489 x 1.0974 x 10^7 m⁻¹)
Solving this equation will give you the wavelength (λ) in meters. To convert it to angstroms (Å), multiply the value by 10^10:
λ (Å) = λ (m) * 10^10
By following these steps and performing the necessary calculations, you can determine the wavelength in Å of the photon absorbed during the hydrogen atom transition from n = 7 to n = 9.
1/L = 10^7 (1/49 - 1/81)
1/L = 10^7( .00806 ) = .806* 10^4
L = 1.24 * 10^-4 meters