Ch.7 - Quantum-Mechanical Model of the Atom

Tro - Chemistry: A Molecular Approach 4th Edition

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Tro 4th Edition

Ch.7 - Quantum-Mechanical Model of the Atom

Problema 72

Capitolo 7, Problema 72

An electron in a hydrogen atom relaxes to the n = 4 level, emitting light of 114 THz. What is the value of n for the level in which the electron originated?

Guida verificata passo dopo passo

Step 1: Understand that the frequency of the light emitted when an electron relaxes from a higher energy level to a lower one in a hydrogen atom can be calculated using the Rydberg formula: $\frac{1}{\lambda} = R_H \left(\frac{1}{n_1^2} - \frac{1}{n_2^2}\right)$, where $\lambda$ is the wavelength of the light, $R_H$ is the Rydberg constant for hydrogen (approximately 1.097 x \$10^7$ m$^{-1}$), $n_1$ is the principal quantum number of the lower energy level, and $n_2$ is the principal quantum number of the higher energy level.

Step 2: Convert the frequency of the light to wavelength using the formula $\lambda = \frac{c}{\nu}$, where $c$ is the speed of light (approximately $3.00 \times 10^8$ m/s) and $\nu$ is the frequency of the light.

Step 3: Substitute the known values into the Rydberg formula and solve for $n_2$. Remember that $n_1$ is 4 (the level to which the electron relaxes) and $\lambda$ is the value you calculated in the previous step.

Step 4: You will get a quadratic equation in terms of $n_2$. Solve this quadratic equation to find the value of $n_2$.

Step 5: The solution to the quadratic equation will give two possible values for $n_2$. However, since $n_2$ must be greater than $n_1$ (because the electron is relaxing to the $n_1$ level), choose the larger of the two solutions as the value of $n_2$.

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Energy Levels in Hydrogen Atom

In a hydrogen atom, electrons occupy discrete energy levels, denoted by quantum numbers (n). The energy levels are quantized, meaning electrons can only exist in specific states. The difference in energy between these levels determines the frequency of light emitted or absorbed when an electron transitions between them.

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Hydrogenation Reactions

Frequency and Energy Relationship

The frequency of light emitted by an electron transitioning between energy levels is directly related to the energy difference between those levels. This relationship is described by the equation E = hν, where E is the energy difference, h is Planck's constant, and ν is the frequency of the emitted light. Thus, knowing the frequency allows us to calculate the energy change associated with the electron's transition.

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Frequency-Wavelength Relationship

Rydberg Formula

The Rydberg formula provides a way to calculate the wavelengths (or frequencies) of light emitted during electron transitions in hydrogen. It is expressed as 1/λ = R_H(1/n1² - 1/n2²), where R_H is the Rydberg constant, n1 and n2 are the principal quantum numbers of the lower and upper energy levels, respectively. This formula is essential for determining the initial energy level (n) from which the electron transitioned.

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