Question

Calculate the de Broglie wavelength for thermal neutrons at a temperature of 100 degrees C

Answer 1

The mass of a neutron is 1.68×10⁻²⁷ kg

T = 100 °C = 373.15 K

Given: The average kinetic energy of a thermal neutron is given by:

(0) K = (3 / 2) k * T,

where
k = the Boltzmann constant = 1.380×10⁻²³
T = the temperature of the containing matter in degrees Kelvin (absolute) = 373.15 K

The simplest form of the de Broglie wavelength equation is:

(1) λ = h / p,

where
h = Planck's constant = 6.63 x 10^-34

(2) p = particle momentum = m * v

The relationship between momentum and kinetic energy is:

(3) K = (1/2) m * v²

Multiplying both sides by m:

(4) m * K = (1/2) * (m * v)²

= (1/2) * p^2

Solving for p:
(5) p = √(2m * K)

Substituting (5) in (1), we get an expression relating de Broglie wavelength to K kinetic energy:
(6) λ = h / √(2m * K)

Substituting (0) in (6),
(7) λ = h / √(2m * 1.5 * k * T)

= 6.63 x 10^-34 / √(2 * 1.68×10⁻²⁷ * 1.5 * 1.380×10⁻²³ * 373.15 )

= 1.30 x 10^-10 = 1.30 Å