To calculate the speed of a neutron that has a de Broglie wavelength equal to the interionic spacing of table salt ($0.282 \, \text{nm}$), we use the de Broglie wavelength formula:

$\lambda = \frac{h}{mv}$

Where:

• $\lambda$ = de Broglie wavelength = $0.282 \, \text{nm} = 0.282 \times 10^{-9} \, \text{m}$
• $h$ = Planck's constant = $6.626 \times 10^{-34} \, \text{J·s}$
• $m$ = mass of a neutron = $1.675 \times 10^{-27} \, \text{kg}$
• $v$ = speed of the neutron (what we need to find)

Rearranging the formula to solve for $v$: $v = \frac{h}{m\lambda}$

Substituting the known values:

$v = \frac{6.626 \times 10^{-34}}{(1.675 \times 10^{-27}) \times (0.282 \times 10^{-9})}$

Perform the calculation:

1. $1.675 \times 10^{-27} \times 0.282 \times 10^{-9} = 4.72515 \times 10^{-37}$
2. $6.626 \times 10^{-34} \div 4.72515 \times 10^{-37} = 1.402 \times 10^{3} \, \text{m/s}$

Convert to km/s:

$v = 1.402 \, \text{km/s}$

Final Answer:

The neutron must have a speed of 1.402 km/s.

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Would you like a detailed explanation of the steps?

Related Questions:

1. What is the significance of the de Broglie wavelength in quantum mechanics?
2. How is the interionic spacing of table salt experimentally determined?
3. What are the typical applications of de Broglie wavelengths in physics?
4. How does the mass of a particle affect its de Broglie wavelength?
5. What is the relationship between de Broglie wavelength and kinetic energy?

Tip:

The de Broglie wavelength decreases as the particle's mass or speed increases, making quantum effects significant only for very small particles like electrons, protons, and neutrons.