To determine the wavelengths that hydrogen atoms absorb to reach the n=8 and n=9 states from the ground state (n=1), you can use the Rydberg formula:
[tex]\[ \frac{1}{\lambda} = R_H \left( \frac{1}{n_1^2} - \frac{1}{n_2^2} \right) \][/tex]
where:
- [tex]\( \lambda \)[/tex] is the wavelength of the absorbed or emitted light,
- [tex]\( R_H \)[/tex] is the Rydberg constant, which is approximately [tex]\( 1.097 \times 10^7 \, \text{m}^{-1} \)[/tex],
- [tex]\( n_1 \)[/tex] is the principal quantum number of the lower energy level (ground state, n=1),
- [tex]\( n_2 \)[/tex] is the principal quantum number of the higher energy level (either n=8 or n=9).
First, calculate the wavelength for the transition to n=8:
1. Use [tex]\( n_1 = 1 \)[/tex] and [tex]\( n_2 = 8 \)[/tex]:
[tex]\[ \frac{1}{\lambda} = R_H \left( \frac{1}{1^2} - \frac{1}{8^2} \right) = 1.097 \times 10^7 \, \text{m}^{-1} \left( 1 - \frac{1}{64} \right) \][/tex]
2. Simplify the expression inside the parentheses:
[tex]\[ 1 - \frac{1}{64} = \frac{64}{64} - \frac{1}{64} = \frac{63}{64} \][/tex]
3. Substitute back into the formula:
[tex]\[ \frac{1}{\lambda} = 1.097 \times 10^7 \, \text{m}^{-1} \times \frac{63}{64} \][/tex]
4. Calculate the reciprocal of the wavelength:
[tex]\[ \frac{1}{\lambda} = 1.079 \times 10^7 \, \text{m}^{-1} \][/tex]
5. Find [tex]\(\lambda\)[/tex] by taking the reciprocal:
[tex]\[ \lambda = \frac{1}{1.079 \times 10^7 \, \text{m}^{-1}} \approx 9.27 \times 10^{-8} \, \text{m} = 92.7 \, \text{nm} \][/tex]
Next, calculate the wavelength for the transition to n=9:
1. Use [tex]\( n_1 = 1 \)[/tex] and [tex]\( n_2 = 9 \)[/tex]:
[tex]\[ \frac{1}{\lambda} = R_H \left( \frac{1}{1^2} - \frac{1}{9^2} \right) = 1.097 \times 10^7 \, \text{m}^{-1} \left( 1 - \frac{1}{81} \right) \][/tex]
2. Simplify the expression inside the parentheses:
[tex]\[ 1 - \frac{1}{81} = \frac{81}{81} - \frac{1}{81} = \frac{80}{81} \][/tex]
3. Substitute back into the formula:
[tex]\[ \frac{1}{\lambda} = 1.097 \times 10^7 \, \text{m}^{-1} \times \frac{80}{81} \][/tex]
4. Calculate the reciprocal of the wavelength:
[tex]\[ \frac{1}{\lambda} = 1.083 \times 10^7 \, \text{m}^{-1} \][/tex]
5. Find [tex]\(\lambda\)[/tex] by taking the reciprocal:
[tex]\[ \lambda = \frac{1}{1.083 \times 10^7 \, \text{m}^{-1}} \approx 9.23 \times 10^{-8} \, \text{m} = 92.3 \, \text{nm} \][/tex]
To classify these wavelengths in the electromagnetic spectrum:
- The calculated wavelengths (92.7 nm and 92.3 nm) fall within the ultraviolet (UV) range of the electromagnetic spectrum, as UV light generally ranges from about 10 nm to 400 nm.
Hence, the photons that hydrogen atoms absorb to reach the n=8 and n=9 states from the ground state lie in the ultraviolet region of the electromagnetic spectrum.
