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.
To calculate the value of n for the higher energy Bohr orbit involved in the emission of this light, we can use the Rydberg equation:
1/λ = R*(1/n₁² - 1/n₂²)
Where:
- λ is the wavelength of the line (656 nm)
- R is the Rydberg constant (approximately 1.097 × 10^7 m⁻¹)
- n₁ is the value for the lower energy orbit (2)
- n₂ is the value we want to calculate for the higher energy orbit
First, we need to convert the wavelength from nanometers to meters:
656 nm = 656 × 10⁻⁹ m
Now, we can substitute the values into the Rydberg equation:
1/(656 × 10⁻⁹ m) = (1.097 × 10^7 m⁻¹) * (1/2² - 1/n₂²)
To isolate n₂, we can rearrange the equation:
1/n₂² = 1/(2²) - (1/(656 × 10⁻⁹ m) * (1.097 × 10^7 m⁻¹))
Now, simplify the equation:
1/n₂² = 1/4 - (1.097 × 10⁷)/(656 × 10⁻⁹)
1/n₂² = 1/4 - 1.674 × 10⁷
Next, compute the right-hand side:
1/n₂² = (1 - 4*(1.674 × 10⁷))/4
1/n₂² = (1 - 6.696 × 10⁷)/4
Finally, solve for n₂:
n₂² = 4/(1 - 6.696 × 10⁷)
n₂² = 4/(-6.696 × 10⁷)
n₂² = -5.970 × 10⁻⁸
Since the value of n must be a positive integer, we disregard the negative result. Therefore, the value of n for the higher energy Bohr orbit involved in the emission of this light is not well-defined based on the given information.
1/wavelength = R*(1/4 - 1/x^2)
Convert wavelength to meters.
You know R, the 1/4 is 1/2^2. X will be the higher orbit.