For He+, a transition takes place from the orbit of radius 105.8 pm to the orbit of radius 26.45 pm. The wavelength (in nm) of the emitted photon during the transition is ___.
[Use: Bohr radius, a = 52.9 pm, Rydberg constant, 𝑅H = 2.2 × 10−18 J, Planck’s constant, h = 6.6 × 10−34 J s, Speed of light, c = 3 × 108 m s−1]
[Use: Bohr radius, a = 52.9 pm, Rydberg constant, 𝑅H = 2.2 × 10−18 J, Planck’s constant, h = 6.6 × 10−34 J s, Speed of light, c = 3 × 108 m s−1]
Correct Answer: 30
Solution and Explanation
Photon Wavelength Calculation
Given: The formula for the radius \( r \) is:
\(r = 52.9 \times \frac{n^2}{z} \,\, \text{pm}\)
From the equation:
\(\therefore 105.8 = \frac{52.9 \times n^2}{2} \,\,\, \therefore n_2 = 2\)
Similarly, for \( n_1 \):
\(26.45 = 52.9 \times \frac{n^2}{2} \,\,\, \therefore n_1 = 1\)
Energy Change Formula:
The energy change \( \Delta E \) is given by:
\(\Delta E = R_H h C \times z^2 \left[ \frac{1}{n_1^2} - \frac{1}{n_2^2} \right]\)
Using the equation for wavelength, we have:
\(\frac{hc}{\lambda} = R_H h C \times z^2 \left[ \frac{1}{n_1^2} - \frac{1}{n_2^2} \right]\)
Substituting the known values:
\(\frac{6.6 \times 10^{-34} \times 3 \times 10^8}{\lambda} = 2.2 \times 10^{-18} \times 4 \times \frac{3}{4}\)
Solving for \( \lambda \):
\(\therefore \lambda = 300 \, \text{Å}\)
Finally, converting to nanometers:
\(\therefore \lambda = 30 \, \text{nm}\)
The wavelength of the emitted photon is 30 nm.
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Concepts Used:
Solutions
A solution is a homogeneous mixture of two or more components in which the particle size is smaller than 1 nm.
For example, salt and sugar is a good illustration of a solution. A solution can be categorized into several components.
Types of Solutions:
The solutions can be classified into three types:
- Solid Solutions - In these solutions, the solvent is in a Solid-state.
- Liquid Solutions- In these solutions, the solvent is in a Liquid state.
- Gaseous Solutions - In these solutions, the solvent is in a Gaseous state.
On the basis of the amount of solute dissolved in a solvent, solutions are divided into the following types:
- Unsaturated Solution- A solution in which more solute can be dissolved without raising the temperature of the solution is known as an unsaturated solution.
- Saturated Solution- A solution in which no solute can be dissolved after reaching a certain amount of temperature is known as an unsaturated saturated solution.
- Supersaturated Solution- A solution that contains more solute than the maximum amount at a certain temperature is known as a supersaturated solution.