Question

The quantum-mechanical treatment of the \(\mathrm{H}\) atom gives the energy, \(E\), of the electron as a function of \(n:\) $$E=-\frac{h^{2}}{8 \pi^{2} m_{e} a_{0}^{2} n^{2}} \quad(n=1,2,3, \ldots) $$where \(h\) is Planck's constant, \(m_{e}\) is the electron's mass, and \(a_{0}\) is \(52.92 \times 10^{-12} \mathrm{~m}\) (a) Write the expression in the form \(E=-\) (constant) \(\left(1 / n^{2}\right),\) evaluate the constant (in \(J\) ), and compare it with the corresponding expression from Bohr's theory. (b) Use the expression from part (a) to find \(\Delta E\) between \(n=2\) and \(n=3\) (c) Calculate the wavelength of the photon that corresponds to this energy change.

Short answer

The constant is approximately \(2.18 \times 10^{-18}\) J. \( \Delta E \) between \( n=2 \) and \( n=3 \) is calculated using energy levels. The corresponding photon wavelength is determined using \( \lambda = \frac{hc}{\Delta E} \).

Step-by-step solution

- Rewriting the Energy Expression

Given the energy expression from quantum mechanics: \[ E = -\frac{h^{2}}{8 \pi^{2} m_{e} a_{0}^{2} n^{2}} \] We need to rewrite it in the form: \[ E = -\text{(constant)} \left(\frac{1}{n^{2}}\right) \] To do this, extract the constant from the expression: \[ \text{constant} = \frac{h^{2}}{8 \pi^{2} m_{e} a_{0}^{2}} \]

- Evaluating the Constant

To evaluate the constant, insert the values of Planck's constant (\(h\)), the electron mass (\(m_e\)), and \(a_0\): \(h = 6.626 \times 10^{-34} \) Js, \(m_e = 9.109 \times 10^{-31} \) kg, \(a_0 = 52.92 \times 10^{-12} \) m.\[ \text{constant} = \frac{(6.626 \times 10^{-34})^2}{8 \pi^{2} (9.109 \times 10^{-31}) (52.92 \times 10^{-12})^2} \] Evaluate the expression to find the constant in joules.

- Comparing with Bohr's Theory

The Bohr model gives the energy as: \[ E = -2.18 \times 10^{-18} \cdot \left(\frac{1}{n^{2}}\right) \text{ J} \] Comparing this to the evaluated constant should show agreement between the quantum mechanical treatment and Bohr's theory.

- Finding \(\Delta E\) between \(n=2\) and \(n=3\)

Using the expression derived, calculate the energy difference \(\Delta E\) between \(n = 2\) and \(n = 3\):\[ \Delta E = E_3 - E_2 \] Expressed as \[ \Delta E = - \frac{constant}{3^2} - \left( - \frac{constant}{2^2} \right) \] Substitute the constant into this equation to find the energy difference.

- Calculating the Wavelength of the Photon

The energy change \( \Delta E \) corresponds to the energy of a photon, which can be related to its wavelength \( \lambda \) by: \[ \Delta E = \frac{hc}{\lambda} \]Solving for wavelength \( \lambda \), we get: \[ \lambda = \frac{hc}{\Delta E} \] Insert the values of Planck's constant \(h\), the speed of light \(c = 3.00 \times 10^8 \) m/s, and the calculated \( \Delta E \) to find the wavelength.

Key concepts

Energy Levels

In the quantum mechanics of hydrogen atom, the energy of an electron is quantized and depends upon the principal quantum number, denoted as \(n\). This quantization means that the electron can only inhabit specific energy levels, which are discrete and not continuous. The formula for the energy of these levels is given by \[ E = - \frac{h^2}{8 \pi^2 m_e a_0^2 n^2} \] where every variable has a specific constant value:
\( h \) is Planck's constant, \( m_e \) is the electron's mass, and \( a_0 \) is the Bohr radius.
The negative sign indicates that the energy is bound within the atom, and greater absolute values of \(n\) represent higher energy states. As \(n\) increases, the energy levels get closer together and the electron becomes less tightly bound.

Planck's Constant

Planck's constant, denoted by \(h\), is a fundamental quantity in quantum mechanics and plays a crucial role in determining the energy levels of an atom. Its value is approximately \( 6.626 \times 10^{-34}\) joule seconds (Js). This constant is pivotal in quantizing various physical properties such as the energy of electrons and photons. For hydrogen atoms, Planck's constant appears in the energy expression:
\[ E = - \frac{h^2}{8 \pi^2 m_e a_0^2 n^2} \]
By measuring energy transitions and the associated photon emissions, Planck's constant helps in relating the energy levels to the emitted or absorbed light. Hence, it serves as a bridge connecting the quantum world with the observable spectral lines.

Bohr Model

The Bohr model, introduced by Niels Bohr, was an early attempt to describe the hydrogen atom. Although it has been superseded by more advanced quantum mechanics, the Bohr model provides a reasonably accurate description of the atom's energy levels:
\[ E = -2.18 \times 10^{-18} \( \frac{1}{n^2} \) \]This model describes electrons orbiting the nucleus in discrete orbits. Each orbit corresponds to a specific energy level. Transitions between these orbits result in the emission or absorption of photons. The quantum mechanical model expands on Bohr’s concepts, refining the energy calculations and explaining phenomena like electron wavefunctions and orbital shapes. While the Bohr model is simpler, it lays the groundwork for understanding quantum mechanics.

Photon Wavelength

When an electron transitions between energy levels in a hydrogen atom, the energy difference is emitted or absorbed as a photon. The wavelength of this photon can be calculated using the formula:
\[ \lambda = \frac{h c}{\Delta E} \]where \( h \) is Planck's constant and \( c \) is the speed of light. The energy difference \( \Delta E \) is the difference between the initial and final energy levels. For example, the energy transition between \( n = 2 \) and \( n = 3 \) results in a photon with a specific wavelength. By measuring the wavelength, one can infer the energy levels involved in the transition. Photon wavelength is thus crucial in spectroscopy, enabling scientists to study the atomic structure and electron configurations.

Energy Transitions

Energy transitions in the hydrogen atom occur when an electron jumps from one energy level to another. According to quantum mechanics, these transitions only occur between discrete levels, each associated with a specific quantum number \( n \). The energy difference between two such levels is given by:
\[ \Delta E = E_f - E_i = -\frac{h^2}{8 \pi^2 m_e a_0^2} \left( \frac{1}{n_f^2} - \frac{1}{n_i^2} \right) \]
where \( E_f \) and \( E_i \) are the final and initial energies, respectively. This energy difference is often emitted or absorbed as a photon. Understanding these transitions allows us to explain phenomena such as spectral lines, which are particular wavelengths of light associated with specific transitions. Each transition’s energy corresponds to a specific wavelength of light, revealing details about the atom's structure and behavior.