Question

Calculate the velocity and kinetic energy of an electron having wavelength of 0.21nm.

Answer 1

As we know that: K.E. =12mv2 (i)and λ=hP=hmvor v=hmλputting the above expression of v in equation (i) we'll get:K.E. = 12mhmλ2=12h2mλ2Putting the values of h, m and λ, the K.E. can be calculated as:K.E. = 126.626×10-34 J s29.11×10-31 Kg 4.8×10-12 m2K.E. = 4.39×10-674.20×10-53=1.05×10-14 J

Answer 2

The velocity of the electron is 3.342 × 10⁻¹⁰ m/s and the Kinetic Energy of the electron is 5.082 × 10⁻²⁰ J

According to the De Broglie wavelength,

λ = h/mv , Where, λ = wavelength of the electron, h = Planck's constant, m = mass of the electron, v = velocity of the electron.

And we know, the Momentum, P = mv = $\sqrt{2mK.E.}$ , K.E. = the kinetic energy of the electron

∴ λ = h/$\sqrt{2mK.E.}$

Give, the wavelength of the electron is (λ) = 0.21 nm = 0.21 × 10⁻⁹ m

∴ K.E. = h²/2mλ²

⇒  K.E. = (6.6 × 10⁻³⁴)² / 2 × 9.1 × 10⁻³¹× (10⁻⁹)²

⇒ K.E. = 0.5082 × 10⁻¹⁹ J

⇒ K.E. = 5.082 × 10⁻²⁰ J

So, the Kinetic energy is 5.082 × 10⁻²⁰ J

K.E. = $\frac{1}{2} mv^2$

Velocity (v) = $\sqrt{\frac{2K.E.}{m} }$

⇒ v = $\sqrt{\frac{2* 5.082 * 10^{-20} }{9.1*10^{-31} } }$

⇒ v = 3.342 × 10⁻¹⁰ m/s

So, the velocity of the electron is 3.342 × 10⁻¹⁰ m/s