Question

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Thermal neutrons are neutrons that move at speeds comparable to those of air molecules at room temperature. These neutrons are most effective in initiating a nuclear chain reaction among 235U isotopes. Calculate the wavelength (in nm) associated with a beam of neutrons moving at 7.00 102 m/s. (Mass of a neutron = 1.675 10-27 kg.)

Answer 1

0.565 nm (3 s.f.)

Step-by-step explanation:

De Broglie Wavelength Formula

`\lambda=(h)/(p)=(h)/(mv)`

where:

• λ = the de Broglie wavelength (m)
• h = Planck's constant (J s)
• p = momentum of the particle (kg m/s)
• m = mass of the particle (kg)
• v = speed of the particle (m/s)

Planck's Constant

A constant relating the energy of a photon to its frequency:

`\sf h = 6.6261 * 10^(-34) J\:s`

Given:

• v = 7.00 × 10² m/s
• m = 1.675 × 10⁻²⁷ kg

Substitute the given values into the formula (along with Planck's Constant):

`\implies \lambda=\sf (6.6261 * 10^(-34))/((1.675 * 10^(-27))(7.00 * 10^2))`

`\implies \lambda=\sf (6.6261 * 10^(-34))/(1.675 * 7.00* 10^(-27)* 10^2)`

`\implies \lambda=\sf (6.6261 * 10^(-34))/(1.675 * 7.00* 10^(-27+2))`

`\implies \lambda=\sf (6.6261 * 10^(-34))/(11.725* 10^(-25))`

`\implies \lambda=\sf (6.6261)/(11.725)* (10^(-34))/(10^(-25))`

`\implies \lambda=\sf 0.5651257...* (10^(-34))/(10^(-25))`

`\implies \lambda=\sf 0.5651257...* 10^(-34-(-25))`

`\implies \lambda=\sf 0.5651257...* 10^(-9)`

`\implies \lambda=\sf 5.651257...* 10^(-10)\:\:m`

To convert meters (m) to nanometers (nm), multiply by 10⁹:

`\implies \lambda=\sf (5.651257...* 10^(-10) * 10^9)\:\:nm`

`\implies \lambda=\sf (5.651257...* 10^(-10+9))\:\:nm`

`\implies \lambda=\sf (5.651257...* 10^(-1))\:\:nm`

`\implies \lambda=\sf 0.5651257...\:\:nm`

`\implies \lambda=\sf 0.565\:\:nm\:\:(3\:s.f.)`

Exponent rules used

`\textsf{Product Rule}: \quad \sf a^b \cdot a^c=a^(b+c)`

`\textsf{Quotient Rule}: \quad \sf (a^b)/(a^c)=a^(b-c)`