1. The Rydberg Formula
The Rydberg formula calculates the energy change for electronic transitions in hydrogen:
```
1/λ = R (1/n₁² - 1/n₂²)
```
Where:
* λ is the wavelength of the emitted or absorbed light
* R is the Rydberg constant (1.097 x 10⁷ m⁻¹)
* n₁ is the initial energy level (lower energy level)
* n₂ is the final energy level (higher energy level)
2. Calculate the Wavelength (λ)
* n₁ = 2 (initial level)
* n₂ = 6 (final level)
Plug these values into the Rydberg formula:
```
1/λ = (1.097 x 10⁷ m⁻¹) (1/2² - 1/6²)
1/λ = 2.438 x 10⁶ m⁻¹
λ = 4.10 x 10⁻⁷ m
```
3. Calculate Energy (ΔE)
We can use the following relationship to relate wavelength and energy:
```
ΔE = hc/λ
```
Where:
* h is Planck's constant (6.626 x 10⁻³⁴ J⋅s)
* c is the speed of light (2.998 x 10⁸ m/s)
* λ is the wavelength (calculated above)
Plug in the values:
```
ΔE = (6.626 x 10⁻³⁴ J⋅s) (2.998 x 10⁸ m/s) / (4.10 x 10⁻⁷ m)
ΔE = 4.84 x 10⁻¹⁹ J
```
4. Convert to kJ/mol
* Convert J to kJ: Divide by 1000
* Convert per atom to per mole: Multiply by Avogadro's number (6.022 x 10²³ atoms/mol)
```
ΔE = (4.84 x 10⁻¹⁹ J) * (1 kJ/1000 J) * (6.022 x 10²³ atoms/mol)
ΔE ≈ 291 kJ/mol
```
Therefore, the change in energy (ΔE) for the electron transition from n=6 to n=2 in a hydrogen atom is approximately 291 kJ/mol. This is a positive value, indicating that energy is absorbed during this transition.
