The wavelength of light from the spectral emission line of sodium is 589 nm. Find the kinetic energy at which (a) an electron, and (b) a neutron, would have the same de Broglie wavelength.

To find the kinetic energy at which an electron and a neutron would have the same de Broglie wavelength as the sodium light at 589 nm, we first need to understand the de Broglie wavelength formula. This formula relates the wavelength of a particle to its momentum, which is given by:

De Broglie Wavelength Formula

The de Broglie wavelength (\(\lambda\)) is expressed as:

\(\lambda = \frac{h}{p}\)

where:

• \(h\) is Planck's constant (\(6.626 \times 10^{-34} \, \text{Js}\))
• \(p\) is the momentum of the particle, which can be calculated as \(p = mv\), where \(m\) is mass and \(v\) is velocity.

Finding Kinetic Energy

The kinetic energy (KE) of a particle is given by the formula:

\(KE = \frac{1}{2} mv^2\)

We can relate kinetic energy to momentum using the equation:

\(p = \sqrt{2m \cdot KE}\)

Substituting this into the de Broglie wavelength formula gives:

\(\lambda = \frac{h}{\sqrt{2m \cdot KE}}\)

Rearranging this equation allows us to solve for kinetic energy:

\(KE = \frac{h^2}{2m\lambda^2}\)

Calculating for the Electron

First, let’s calculate the kinetic energy for the electron:

• Mass of electron (\(m_e\)): \(9.11 \times 10^{-31} \, \text{kg}\)
• Wavelength (\(\lambda\)): \(589 \, \text{nm} = 589 \times 10^{-9} \, \text{m}\)

Now, substituting these values into the kinetic energy formula:

\(KE_e = \frac{(6.626 \times 10^{-34})^2}{2 \cdot (9.11 \times 10^{-31}) \cdot (589 \times 10^{-9})^2}\)

Calculating this gives:

\(KE_e \approx 1.24 \times 10^{-18} \, \text{J}\)

Calculating for the Neutron

Next, we perform the same calculation for the neutron:

• Mass of neutron (\(m_n\)): \(1.675 \times 10^{-27} \, \text{kg}\)

Using the same wavelength, we substitute into the kinetic energy formula:

\(KE_n = \frac{(6.626 \times 10^{-34})^2}{2 \cdot (1.675 \times 10^{-27}) \cdot (589 \times 10^{-9})^2}\)

Calculating this gives:

\(KE_n \approx 2.57 \times 10^{-14} \, \text{J}\)

Summary of Results

To summarize, the kinetic energies required for an electron and a neutron to have the same de Broglie wavelength of 589 nm are:

• Electron: \(1.24 \times 10^{-18} \, \text{J}\)
• Neutron: \(2.57 \times 10^{-14} \, \text{J}\)

This demonstrates how the mass of the particle significantly influences the kinetic energy required to achieve the same wavelength. The heavier neutron requires much more energy compared to the lighter electron to reach the same de Broglie wavelength.