An astrophysicist working at an observatory is interested in finding clouds of hydrogen in the galaxy. Usually hydrogen is detected by looking for the Balmer series of spectral lines in the visible spectrum. Unfortunately, the instrument that detects hydrogen emission spectra at this particular observatory is not working very well and only detects spectra in the infrared region of electromagnetic radiation. Therefore the astrophysicist decides to check for hydrogen by looking at the Paschen series, which produces spectral lines in the infrared part of the spectrum. The Paschen series describes the wavelengths of light emitted by the decay of electrons from higher orbits to the n=3 level.

What wavelength λ should the astrophysicist look for to detect a transition of an electron from the n=5 to the n=3 level?

(1/wavelength) = R(1/9 - 1/25)

The 9 is 3^2 and the 25 is 5^2.
R = 1.0973E7

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To find the wavelength λ corresponding to the transition of an electron from the n=5 to the n=3 level in the Paschen series, we can use the Rydberg formula:

1/λ = R * (1/n1^2 - 1/n2^2)

In this formula, R is the Rydberg constant (which is 1.097 × 10^7 m^-1), n1 is the initial energy level (in this case n=5), and n2 is the final energy level (in this case n=3).

Plugging in the values into the formula, we get:

1/λ = (1.097 × 10^7 m^-1) * (1/5^2 - 1/3^2)

1/λ = (1.097 × 10^7 m^-1) * (1/25 - 1/9)

1/λ = (1.097 × 10^7 m^-1) * (8/225)

1/λ = 8.76 × 10^4 m^-1

To obtain the wavelength λ, we can take the reciprocal of both sides:

λ = 1 / (8.76 × 10^4 m^-1)

λ ≈ 1.14 × 10^-5 m

Therefore, the astrophysicist should look for a wavelength of approximately 1.14 × 10^-5 meters to detect the transition of an electron from the n=5 to the n=3 level in the Paschen series.

To find the wavelength that the astrophysicist should look for to detect a transition of an electron from the n=5 to the n=3 level in the Paschen series, we can use the formula for the wavelengths of the emission lines in the Paschen series.

The formula is given by:

1/λ = R (1/n1^2 - 1/n2^2)

where λ is the wavelength, R is the Rydberg constant (which is 1.097 × 10^7 m^-1), and n1 and n2 are the principal quantum numbers representing the initial and final energy levels of the electron transition, respectively.

In this case, n1 = 5 and n2 = 3. Substituting these values into the formula:

1/λ = R (1/5^2 - 1/3^2)

Simplifying the equation:

1/λ = R (1/25 - 1/9)

1/λ = R ((9 - 25) / (9 * 25))

1/λ = R (-16 / 225)

1/λ = -R/225

Now, we can find the value of λ by taking the reciprocal of both sides:

λ = 225 / R

Substituting the value of R:

λ = 225 / (1.097 × 10^7)

Calculating this expression gives:

λ ≈ 2.046 × 10^-6 meters (or 2.046 micrometers)

So, the astrophysicist should look for a wavelength of approximately 2.046 micrometers to detect the transition of an electron from the n=5 to the n=3 level in the Paschen series.