The wavelength of a neutron with a translatory kinetic energy equal to kT at 300 K isa)178 pmb)200 pmc)17.8 pmd)20.0 pmCorrect answer is option 'A'. Can you explain this answer?

Question

Answer

Kinetic energy = kT

Answer

Answer

The wavelength of a neutron with a translatory kinetic energy equal to kT at 300 K can be calculated using the de Broglie wavelength equation. The de Broglie wavelength (λ) is given by the equation:

λ = h / p

Where:
λ is the wavelength
h is the Planck's constant (6.626 x 10^-34 J·s)
p is the momentum of the particle

To calculate the momentum of the neutron, we can use the equation:

p = √(2mE)

Where:
p is the momentum
m is the mass of the neutron (1.675 x 10^-27 kg)
E is the kinetic energy of the neutron

Now, let's substitute the values into the equation:

p = √(2mE)
p = √(2 * 1.675 x 10^-27 kg * kT)

Since kT at 300 K is equal to 4.11 x 10^-21 J (Boltzmann constant k = 1.38 x 10^-23 J/K), we can substitute this value:

p = √(2 * 1.675 x 10^-27 kg * 4.11 x 10^-21 J)
p ≈ 4.13 x 10^-24 kg·m/s

Now, we can substitute the momentum value into the de Broglie wavelength equation:

λ = h / p
λ = (6.626 x 10^-34 J·s) / (4.13 x 10^-24 kg·m/s)
λ ≈ 1.60 x 10^-10 m

To convert the wavelength to picometers (pm), we multiply it by 10^12:

λ = 1.60 x 10^-10 m * 10^12 pm/m
λ ≈ 1.60 x 10^2 pm

Therefore, the wavelength of a neutron with a translatory kinetic energy equal to kT at 300 K is approximately 160 pm.

Since none of the answer options match this value, it seems there may be an error in the options provided. However, option 'A' (178 pm) is the closest value to the calculated wavelength.

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