Suppose a sample of hydrogen gas is excited to the n=5 level. As the atoms return to the ground state (Balmer series), they emit light.
a. How many lines are there in the spectrum?
b. Draw an energy-level diagram indicating theses transitions.
c. Calculate the three lowest-energy transitions, and indicate them on your drawing
Same formulas, same process as earlier problems. Post your work if you get stuck.
To answer these questions, we need to understand the energy levels and transitions in the hydrogen atom, specifically the Balmer series.
a. The number of lines in the hydrogen spectrum can be determined using the formula for the Balmer series:
1/λ = R_H (1/2^2 - 1/n^2),
where λ is the wavelength of the emitted light, R_H is the Rydberg constant (1.097 × 10^7 m^-1), and n is the principal quantum number of the energy level.
For the Balmer series, the principal quantum number n ranges from 3 to infinity, but in this case, it is excited to the n=5 level and will transition back to n=2 (ground state). Therefore, we need to find the number of transitions from n=5 to n=2.
Starting from n=5 and transitioning to n=2, we can calculate the number of transitions using the equation:
Number of transitions = n_final - n_initial = 2 - 5 = -3.
The negative sign indicates that the transitions are from higher to lower energy levels. So, in this case, there are 3 transitions in the Balmer series.
b. To draw an energy-level diagram indicating these transitions, we will represent the energy levels as horizontal lines and indicate the transitions with arrows going downward.
```
n = 5 |
n = 4 |
n = 3 |
n = 2 |
n = 1 |
```
c. To calculate the three lowest-energy transitions, we will use the Balmer formula. Taking n=5 as the initial level and n=2 as the final level, we can calculate the wavelength using:
1/λ = R_H (1/2^2 - 1/n^2).
1/λ = R_H (1/2^2 - 1/5^2).
1/λ = R_H (1/4 - 1/25).
1/λ = R_H (25/100 - 4/100).
1/λ = R_H (21/100).
λ = 100/(21 * R_H).
Using the given value for R_H, we can calculate the wavelength for each transition.
For the three lowest-energy transitions (n=5 to n=2), the wavelengths are:
λ_1 = 100/(21 * R_H).
λ_2 = 100/(21 * R_H).
λ_3 = 100/(21 * R_H).
You can substitute the value of R_H to get the actual wavelengths.
Now, mark these three transitions on your energy-level diagram as arrows going downward from n=5 to n=2.