If a \(2.5\) electron-volt photon liberates an electron from silicon with a \(1.1 \mathrm{eV}\) band gap, how much kinetic energy does the emerging electron have? Express in both eV and Joules, and then determine the velocity of the electron if the electron mass is \(9 \times 10^{-31} \mathrm{~kg}\).

Photon energy is the energy carried by a single photon, which can be calculated by the equation \(E = h u \) where \(h\) is Planck's constant and \(u\) is the frequency of the photon. In the context of liberating electrons from material like silicon, the energy of a photon must be greater than the material's bandgap to excite an electron out of its bound state.

For instance, if a photon has an energy of 2.5 electron-volts (eV), it possesses a certain quantum of energy, which can be transferred to an electron during interactions such as absorption. The concept is crucial as it underpins the photoelectric effect, where electrons are ejected from materials when illuminated by light of sufficient energy.

The bandgap of silicon is a fundamental property of this semiconductor material, referring to the energy difference between its highest valence band and lowest conduction band. For silicon, this bandgap is approximately 1.1 eV. Electrons must gain this minimum energy to jump from the valence band to the conduction band and become free to conduct electric current.

Analogous to a gate that requires a certain effort to be opened, the bandgap restricts the flow of electrons until they receive enough energy from an external source, like a photon. This property is why silicon is used widely in electronic devices, particularly in photovoltaic cells for solar energy conversion, as it can convert the energy from photons into electrical energy by freeing electrons.

The electron-volt (eV) is a unit of energy commonly used in the context of atomic and particle physics, which represents the amount of energy one electron acquires when it moves through a potential difference of one volt. To interconvert between electron-volts and the SI unit of energy, joules (J), we use the conversion factor \(1 eV = 1.6 \times 10^{-19} J\).

This conversion is useful when we need to apply classical physics equations, which typically use SI units. For instance, when calculating the kinetic energy of an electron in joules from its value in electron-volts, as required in the exercise, this conversion factor is essential in bridging the gap between the microscopic scale of atomic interactions and the macroscopic world of classical mechanics.

Determining the velocity of an electron involves using its kinetic energy and mass through the classical kinetic energy formula \(KE = 0.5 \times m \times v^2\). Here, \(KE\) represents kinetic energy, \(m\) is the mass of the electron, and \(v\) is the velocity of the electron. Once we know the kinetic energy in joules, we can solve for velocity \(v\) by rearranging the formula to \(v = \sqrt{2 \times KE / m}\).

As electrons are subatomic particles with a very small mass, their velocities can be quite high even with small amounts of kinetic energy. Calculating the velocity of electrons is fundamental in fields like solid-state physics and electronics, where understanding electron behavior under different energy conditions is essential for the design of various electronic components.