Math Problem Statement

. Calculate the wavelength, in nanometers, of the light emitted by a hydrogen atom when its electron transitions from the n = 8 to the n = 5 principal energy level. A. 3740 nm B. 984 nm C. 148 nm D. 24451 nm E. 1321 nm

Solution

To calculate the wavelength of light emitted during an electron transition in a hydrogen atom, we use the Rydberg formula:

1λ=RH(1nf21ni2)\frac{1}{\lambda} = R_H \left( \frac{1}{n_f^2} - \frac{1}{n_i^2} \right)

Where:

  • λ\lambda is the wavelength of the emitted light,
  • RHR_H is the Rydberg constant, RH=1.097×107m1R_H = 1.097 \times 10^7 \, \text{m}^{-1},
  • nin_i is the initial energy level (in this case ni=8n_i = 8),
  • nfn_f is the final energy level (in this case nf=5n_f = 5).

Step 1: Calculate the wavelength

We can substitute the values into the equation:

1λ=1.097×107m1(152182)\frac{1}{\lambda} = 1.097 \times 10^7 \, \text{m}^{-1} \left( \frac{1}{5^2} - \frac{1}{8^2} \right)

1λ=1.097×107m1(125164)\frac{1}{\lambda} = 1.097 \times 10^7 \, \text{m}^{-1} \left( \frac{1}{25} - \frac{1}{64} \right)

Step 2: Simplify the expression

First, find a common denominator:

125164=64251600=391600\frac{1}{25} - \frac{1}{64} = \frac{64 - 25}{1600} = \frac{39}{1600}

So, the equation becomes:

1λ=1.097×107×391600\frac{1}{\lambda} = 1.097 \times 10^7 \times \frac{39}{1600}

1λ=2.673×105m1\frac{1}{\lambda} = 2.673 \times 10^5 \, \text{m}^{-1}

Step 3: Calculate the wavelength

λ=12.673×105m\lambda = \frac{1}{2.673 \times 10^5} \, \text{m}

λ=3.74×106m\lambda = 3.74 \times 10^{-6} \, \text{m}

Step 4: Convert to nanometers

Since 1m=109nm1 \, \text{m} = 10^9 \, \text{nm}:

λ=3.74×106m×109nm/m=3740nm\lambda = 3.74 \times 10^{-6} \, \text{m} \times 10^9 \, \text{nm/m} = 3740 \, \text{nm}

Thus, the wavelength is 3740 nm, which corresponds to option A.

Answer: A. 3740 nm

Would you like further clarification or details?


Here are 5 related questions:

  1. How is the energy of a photon related to its wavelength?
  2. What is the significance of the Rydberg constant in quantum physics?
  3. How does the Bohr model of the hydrogen atom explain spectral lines?
  4. How do principal quantum numbers affect energy levels in an atom?
  5. What is the Balmer series, and how does it relate to visible hydrogen emissions?

Tip: When dealing with light transitions, higher energy differences result in shorter wavelengths, which corresponds to ultraviolet or visible light.

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Math Problem Analysis

Mathematical Concepts

Quantum Physics
Atomic Structure
Wavelength Calculation

Formulas

Rydberg formula: 1/λ = R_H (1/n_f^2 - 1/n_i^2)

Theorems

Rydberg's formula for hydrogen spectral lines

Suitable Grade Level

College Level (Introductory Physics or Chemistry)