Hydrogen-like ions are those with only one electron such as He+ and Li2+. The Balmer formula for the transitions in these ions can be written as v=Z^2 cR(1/(n_1^2 )-1/(n_2^2 )). Use this formula to calculate the lowest-energy transition in the Lyman and Balmer series for He+ and Li2+.
So use that formula. Post your work if you get stuck.
To calculate the lowest-energy transition in the Lyman and Balmer series for He+ and Li2+, we will use the Balmer formula:
v = Z^2 * c * R * (1/(n_1^2) - 1/(n_2^2))
Where:
- v is the frequency of the transition
- Z is the atomic number of the hydrogen-like ion (equal to the number of protons)
- c is the speed of light (approximately 3 × 10^8 m/s)
- R is the Rydberg constant (approximately 1.097 × 10^7 m^-1)
- n1 and n2 are the principal quantum numbers representing the initial and final energy levels of the electron
For He+:
Since He+ is a hydrogen-like ion, Z = 2 (atomic number of helium). Let's calculate the lowest-energy transition in the Lyman series and Balmer series separately.
1. Lyman series:
In the Lyman series, the initial energy level (n1) is 1, and the final energy level (n2) is greater than 1. The lowest-energy transition in the Lyman series corresponds to n2 = 2.
Substituting the values into the formula:
v = 2^2 * 3 × 10^8 m/s * 1.097 × 10^7 m^-1 * (1/(1^2) - 1/(2^2))
Simplifying the equation:
v = 4 * 2.191 × 10^15 Hz
v = 8.764 × 10^15 Hz
2. Balmer series:
In the Balmer series, the initial energy level (n1) is 2, and the final energy level (n2) is greater than 2. The lowest-energy transition in the Balmer series corresponds to n2 = 3.
Substituting the values into the formula:
v = 2^2 * 3 × 10^8 m/s * 1.097 × 10^7 m^-1 * (1/(2^2) - 1/(3^2))
Simplifying the equation:
v = 4 * 2.191 × 10^15 Hz * (1/4 - 1/9)
v = 4 * 2.191 × 10^15 Hz * (9/36 - 4/36)
v = 4 * 2.191 × 10^15 Hz * (5/36)
v = 1.220 × 10^15 Hz
For Li2+:
Since Li2+ is a hydrogen-like ion, Z = 3 (atomic number of lithium). Let's calculate the lowest-energy transition in the Lyman series and Balmer series separately.
1. Lyman series:
Substituting the values into the formula:
v = 3^2 * 3 × 10^8 m/s * 1.097 × 10^7 m^-1 * (1/(1^2) - 1/(2^2))
v = 9 * 2.191 × 10^15 Hz = 19.719 × 10^15 Hz
2. Balmer series:
Substituting the values into the formula:
v = 3^2 * 3 × 10^8 m/s * 1.097 × 10^7 m^-1 * (1/(2^2) - 1/(3^2))
v = 9 * 2.191 × 10^15 Hz * (1/4 - 1/9)
v = 9 * 2.191 × 10^15 Hz * (9/36 - 4/36)
v = 9 * 2.191 × 10^15 Hz * (5/36)
v = 2.654 × 10^15 Hz
Therefore, the lowest-energy transition in the Lyman series for He+ is 8.764 × 10^15 Hz, and for Li2+ is 19.719 × 10^15 Hz. The lowest-energy transition in the Balmer series for He+ is 1.220 × 10^15 Hz, and for Li2+ is 2.654 × 10^15 Hz.