A wavelength of 6.771 µm is observed in a hydrogen spectrum for a transition that ends in the nf = 7 level. What was ni for the initial level of the electron?

To determine the initial level of the electron (ni) for the observed wavelength of 6.771 µm in the hydrogen spectrum, we can use the Rydberg formula:

1/λ = R_H * (1/ni^2 - 1/nf^2)

where λ is the wavelength, R_H is the Rydberg constant for hydrogen, ni is the initial level, and nf is the final level.

First, let's convert the wavelength from micrometers (µm) to meters (m):

6.771 µm = 6.771 × 10^-6 m

Next, we need to find the Rydberg constant for hydrogen (R_H). The Rydberg constant is given by:

R_H = 1.0973731568539 × 10^7 m^-1

Now, we can rearrange the Rydberg formula to solve for ni:

1/λ = R_H * (1/ni^2 - 1/nf^2)

1/(6.771 × 10^-6 m) = 1.0973731568539 × 10^7 m^-1 * (1/ni^2 - 1/7^2)

Simplifying the equation further:

1/(6.771 × 10^-6 m) = 1.0973731568539 × 10^7 m^-1 * (1/ni^2 - 1/49)

Now, we can solve for ni. Let's start by simplifying the right side of the equation:

1/ni^2 - 1/49 = (49 - ni^2)/(49 * ni^2)

Substituting this back into the equation:

1/(6.771 × 10^-6 m) = 1.0973731568539 × 10^7 m^-1 * (49 - ni^2)/(49 * ni^2)

Now, we can cross-multiply and solve for ni:

ni^2 * (6.771 × 10^-6 m) * (1.0973731568539 × 10^7 m^-1) = (49 * ni^2) * (1.0973731568539 × 10^7 m^-1) - (49 * 6.771 × 10^-6 m)

Simplifying further:

ni^2 * (6.771 × 10^-6 m) * (1.0973731568539 × 10^7 m^-1) = 49 * ni^2 * (1.0973731568539 × 10^7 m^-1) - (49 * 6.771 × 10^-6 m)

Now, we can cancel out the common terms:

6.771 × 10^-6 m * (1.0973731568539 × 10^7 m^-1) = 49 * (1.0973731568539 × 10^7 m^-1) - (49 * 6.771 × 10^-6 m)

Simplifying further:

6.771 × 10^-6 m = 49 * (1.0973731568539 × 10^7 m^-1) - (49 * 6.771 × 10^-6 m)

Now, we can solve for ni by isolating it on one side of the equation:

49 * (1.0973731568539 × 10^7 m^-1) - (49 * 6.771 × 10^-6 m) = 6.771 × 10^-6 m

49 * (1.0973731568539 × 10^7 m^-1) = 6.771 × 10^-6 m + (49 * 6.771 × 10^-6 m)

49 * (1.0973731568539 × 10^7 m^-1) = 6.771 × 10^-6 m * (1 + 49)

Now, we can calculate ni:

ni = sqrt((6.771 × 10^-6 m * (1 + 49)) / (49 * (1.0973731568539 × 10^7 m^-1)))

Evaluating the expression:

ni ≈ 2.997

Therefore, the initial level of the electron (ni) for the observed wavelength of 6.771 µm in the hydrogen spectrum is approximately 2.997.