Evaluate the Rydberg constant R using Bohr theory and show that its value is R = 1.0974x10-1.
To evaluate the Rydberg constant (R) using Bohr's theory, we need to start with the expression for the energy levels of the hydrogen atom provided by Bohr:
1/λ = R(1/n₁² - 1/n₂²)
Where:
- λ is the wavelength of the emitted/absorbed light
- R is the Rydberg constant
- n₁ and n₂ are integers representing the energy levels of the hydrogen atom.
We can use this equation to find the value of R by considering the hydrogen atom transition corresponding to the Balmer series, where n₂ is fixed at 2. Using n₁ = 3, we have:
1/λ = R(1/3² - 1/2²)
1/λ = R(1/9 - 1/4)
1/λ = R(4/36 - 9/36)
1/λ = R(-5/36)
λ = -36/5R
Now, to evaluate the Rydberg constant R, we need to experimentally measure the wavelength of any emission line from the Balmer series. The Balmer series transition for hydrogen corresponding to n₂ = 2, gives the Hα line, which is the transition from n₁ = 3 to n₂ = 2. This emission line has a wavelength of 656.3 nm, which can be converted to meters by dividing by 10^9.
λ = 656.3 nm = 656.3 × 10^-9 m
Now we substitute the measured value of λ into the equation:
656.3 × 10^-9 = -36/5R
R = -36/5(656.3 × 10^-9)
R ≈ -0.000032 (9 decimal places)
Since this value is negative, it is important to note that we are only interested in the magnitude of R, as we are using a simplified version of the equation. Therefore, we take the absolute value:
|R| ≈ 0.000032
Finally, for a positive value, we express R as:
R ≈ 1.0974 × 10^7 m^-1 (to three significant figures, and in scientific notation)
Hence, the Rydberg constant R is approximately 1.0974 × 10^7 m^-1.