To calculate the speed of an electron that can ionize a hydrogen atom, we can utilize the principles of energy conservation and the relationship between energy and velocity.
First, we need to convert the ionization potential of hydrogen from electron volts (eV) to joules (J). Since 1 eV is equal to 1.6 x 10^(-19) J, we can calculate:
Ionization potential = 13.6 eV * (1.6 x 10^(-19) J/eV) = 2.18 x 10^(-18) J
The ionization potential represents the energy required to remove the electron from the hydrogen atom.
Next, we can apply the principle of energy conservation, equating the ionization potential to the kinetic energy of the electron:
Kinetic energy = Ionization potential
The kinetic energy of an object can be expressed as:
Kinetic energy = (1/2) * mass * velocity^2
For the electron, we know the mass (m) is approximately 9.1 x 10^(-31) kg.
Substituting the values and rearranging the equation, we get:
(1/2) * 9.1 x 10^(-31) kg * velocity^2 = 2.18 x 10^(-18) J
Simplifying the equation further:
velocity^2 = (2 * 2.18 x 10^(-18) J) / (9.1 x 10^(-31) kg)
velocity^2 = 4.76 x 10^(12) m^2/s^2
Taking the square root of both sides, we find:
velocity = √(4.76 x 10^(12) m^2/s^2)
velocity ≈ 2.18 x 10^6 m/s
Therefore, the speed of the electron that can ionize a hydrogen atom is approximately 2.18 x 10^6 m/s.