Question:

A photon with a wavelength of 600 nm strikes a metal surface, causing the emission of an electron. The work function of the metal is 3.2 eV. Assume the photon is completely absorbed by the electron and that the electron is initially at rest.

a) Calculate the energy (in eV) of the photon.

b) Determine the kinetic energy (in eV) of the emitted electron.

c) Calculate the velocity (in m/s) of the emitted electron, given that its mass is 9.11 × 10^(-31) kg.

Answer:

a) The energy (E) of a photon is given by the equation:

E = hc / λ

where h is Planck's constant (6.63 × 10^(-34) J·s), c is the speed of light (3.00 × 10^(8) m/s), and λ is the wavelength.

Converting the wavelength to meters:

λ = 600 nm = 600 × 10^(-9) m

Substituting the values into the equation:

E = (6.63 × 10^(-34) J·s × 3.00 × 10^(8) m/s) / (600 × 10^(-9) m) E ≈ 3.31 × 10^(-19) J

To convert Joules to electron volts (eV), we use the conversion: 1 eV = 1.6 × 10^(-19) J.

Thus, the energy of the photon is approximately:

E = 3.31 × 10^(-19) J / (1.6 × 10^(-19) J/eV) E ≈ 2.07 eV

b) The kinetic energy (KE) of the emitted electron can be calculated using the following equation:

KE = E - Φ

where Φ is the work function of the metal.

Substituting the values:

KE = 2.07 eV - 3.2 eV KE ≈ -1.13 eV

The negative value indicates that the electron loses energy in the process.

c) The kinetic energy can also be expressed in terms of the velocity of the emitted electron. The equation is given by:

KE = (1/2)mv^2

where m is the mass of the electron and v is its velocity.

Rearranging the equation to solve for velocity:

v = √(2KE / m)

Substituting the known values:

v = √(2 × (-1.13 eV) × (1.6 × 10^(-19) J/eV) / (9.11 × 10^(-31) kg) v ≈ √(2 × (-1.13) × (1.6 × 10^(19)) / (9.11 × 10^(-31))) v ≈ √((-2.29 × 10^(19)) / (9.11 × 10^(-31))) v ≈ √(-2.51 × 10^(49)) v is not a real number (the square root of a negative number), indicating that the electron does not have a well-defined velocity.

Explanation:

a) In part (a), we use the equation E = hc / λ to calculate the energy of the photon. By substituting the given values, we find that the energy of the photon is approximately 2.07 eV.

b) Part (b) involves calculating the kinetic energy of the emitted electron. The equation KE = E - Φ is used, where E is the energy of the photon and Φ is the work function of the metal. Substituting the values, we determine that the kinetic energy of the electron is approximately -1.13 eV.

c) In part (c), we express the kinetic energy in terms of the velocity of the electron using the equation KE = (1/2)mv^2. Solving for velocity, we substitute the known values and find that the velocity is not a real number. This implies that the electron does not have a well-defined velocity.

Note: The negative kinetic energy and the lack of well-defined velocity in part (c) indicate that the electron loses energy and is not able to completely overcome the work function of the metal.