Question

An 8.00 -eV photon is absorbed by an electron in the \(n=2\) state of a hydrogen atom. Calculate the final speed of the electron.

Short answer

Answer: The final speed of the electron after absorbing the 8.00 eV photon is approximately \(1.27 \times 10^6 \, \text{m/s}\).

Step-by-step solution

Calculate the initial energy of the electron in the n=2 state

We can determine the electron's initial energy using the formula for energy levels in a hydrogen atom: $$E_n = -\frac{13.6 \,\text{eV}}{n^2}$$ where \(n\) is the energy level number. In this case, \(n=2\). $$E_2 = -\frac{13.6 \,\text{eV}}{2^2} = -3.4 \,\text{eV}$$

Calculate the total energy after the photon is absorbed

When the electron absorbs the 8.00 eV photon, its energy increases by the photon's energy: $$E_{total} = E_2 + E_{photon} = -3.4 \,\text{eV} + 8.00 \,\text{eV} = 4.6 \,\text{eV}$$

Find the new energy level and calculate the energy difference

If the new energy state is in the nth energy level, we have: $$E_n=-\frac{13.6 \,\text{eV}}{n^2} = 4.6 \,\text{eV}$$ Solving for n: $$n^2 = \frac{-13.6 \,\text{eV}}{4.6 \,\text{eV}}$$ However, we find that the resulting \(n^2\) is negative which means that the electron is no longer bound to the atom, and it's in a continuous energy state or free electron.

Calculate the final kinetic energy of the electron

In this case, since the electron is in a continuous energy state, its total energy is equal to its kinetic energy: $$E_{f,k} = E_{total} = 4.6 \,\text{eV}$$

Calculate the final speed of the electron

Now we can calculate the speed of the electron using the kinetic energy formula: $$E_{f,k} = \frac{1}{2} mv^2$$ Where \(m\) is the electron mass, and \(v\) is the final speed. First, we need to convert the kinetic energy from eV to Joules: $$E_{f,k} = 4.6 \,\text{eV} \cdot \frac{1.6 \times 10^{-19} \,\text{J}}{\text{eV}} = 7.36 \times 10^{-19} \,\text{J}$$ Now, we can solve for the final speed: $$v = \sqrt{\frac{2E_{f,k}}{m}}$$ Taking \(m\) as the mass of an electron \(9.11 \times 10^{-31} \,\text{kg}\): $$v = \sqrt{\frac{2(7.36 \times 10^{-19} \,\text{J})}{9.11 \times 10^{-31} \,\text{kg}}} = 1.27 \times 10^6 \,\text{m/s}$$ The final speed of the electron after absorbing the 8.00 eV photon is approximately \(1.27 \times 10^6 \, \text{m/s}\).

Key concepts

Photon Absorption

Photon absorption is a fundamental process in which a photon's energy is transferred to an electron. When a photon hits an electron within an atom, such as a hydrogen atom, it may be absorbed if its energy matches the difference between two energy levels of the electron. This energy excites the electron, moving it to a higher energy state.
When a photon with an energy of 8.00 eV is absorbed by an electron in the hydrogen atom, it provides sufficient energy for the electron to transition out of its bound state, resulting in the electron gaining that energy.
After absorbing the photon, energy from the photon causes an increase in the electron's total energy.

• If the electron absorbs enough energy, it becomes unbound from the atom, resulting in what is known as a free electron state.
• This unbinding occurs because the electron's new energy level exceeds the atomic binding energy, freeing the electron from its atom.

Hydrogen Atom Energy Levels

The hydrogen atom has distinct energy levels where electrons can reside. These levels are quantized, meaning electrons can only occupy specific energy states.
In hydrogen, these energy levels are given by the formula: \[ E_n = -\frac{13.6 \, \text{eV}}{n^2} \] where \( n \) is the principal quantum number.
For the \( n=2 \) level, the energy is \(-3.4 \, \text{eV} \), which represents the electron's bound state within the atom.
When the electron absorbs a photon, and its energy increases to \(4.6 \, \text{eV} \), it surpasses the maximum energy it can possess while still bound to the atom.

• This indicates that the electron has moved beyond the highest quantized level, highlighting its transition into freedom from the atom.
• Understanding these energy levels helps explain the absorption and emission spectra observed in hydrogen and other atoms.

Electron Kinetic Energy

When an electron absorbs enough energy to escape its atomic constraints, its total energy is converted into kinetic energy. The kinetic energy (KE) of an electron is the energy it possesses due to its motion, and it is calculated by: \[ KE = \frac{1}{2} mv^2 \] where \( m \) is the mass of the electron and \( v \) is its velocity.
In this scenario, since the electron absorbs an 8.00 eV photon and now has an energy of 4.6 eV in its free state, this energy becomes entirely kinetic. This is because the energy with which the electron was initially bound is overcome, thus liberating and accelerating the electron.

• This newfound kinetic energy defines how fast the electron travels once it escapes the atom, as calculated through its velocity.
• The relationship between energy and velocity allows physicists to understand and predict the behavior of free electrons.

Atomic Physics

Atomic physics explores the structure of the atom and the interactions between its components. A core aspect is understanding how electrons behave within atoms and how they interact with photons.
By examining electron transitions and photon absorption, atomic physics unravels the complexities of atomic energy levels and electron dynamics.

• It explores phenomena such as the absorption and emission spectra, which provide insights into an atom's energy levels.
• Learning how electrons absorb photons to move between quantized energy levels or escape the atom entirely reveals much about atomic stability and reactivity.

These principles are essential for comprehending broader topics in physics, chemistry, and technology, aiding in the development of practical applications such as lasers and solar cells.