Question

What would be the kinetic energy, in J, of an electron with a wavelength of 0.445 nm, which would be equivalent to the wavelength of electromagnetic radiation in the X-ray region? (The mass of an electron is 9.11 × 10⁻²⁸ g.)

Answer 1

The kinetic energy of the electron is approximately 4.45 × 10^-15 J, assuming that the electron is moving at a velocity of about 1.198 × 10^7 m/s.

Step-by-step explanation:

We can use the formula for the energy of a photon of electromagnetic radiation:

E = hc/λ

where h is Planck's constant (6.626 × 10^-34 J·s), c is the speed of light (2.998 × 10^8 m/s), and λ is the wavelength of the radiation.

Since the wavelength of the electron in this question is equivalent to the wavelength of X-ray radiation, we can assume that the energy of the electron is equal to the energy of a photon of X-ray radiation with the same wavelength.

So, we can calculate the energy of the photon:

E = hc/λ = (6.626 × 10^-34 J·s × 2.998 × 10^8 m/s)/(0.445 × 10^-9 m) ≈ 4.45 × 10^-15 J

Since the electron has the same energy as the photon, its kinetic energy is also approximately 4.45 × 10^-15 J.

To convert the mass of the electron from grams to kilograms, we divide by 1000:

mass of electron = 9.11 × 10^-28 kg

Using the formula for kinetic energy:

KE = (1/2)mv^2

where m is the mass of the electron and v is its velocity, we can solve for the velocity of the electron:

KE = (1/2)mv^2

v^2 = (2KE)/m

v = √((2KE)/m)

Substituting the values we have calculated, we get:

√((2KE)/m) = √((2 × 4.45 × 10^-15 J)/(9.11 × 10^-28 kg)) ≈ 1.198 × 10^7 m/s