The problem asks to calculate the speed of different particles if they had the same wavelength as a photon of orange light, which has a wavelength of $\lambda = 625.0 \, \text{nm}$.

To find the speed of each particle, we use de Broglie's wavelength equation:

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

Where:

• $\lambda$ is the wavelength,
• $h$ is Planck's constant ($6.626 \times 10^{-34} \, \text{J} \cdot \text{s}$),
• $m$ is the mass of the particle (in kilograms),
• $v$ is the speed of the particle.

Rearranging the equation to solve for $v$:

$v = \frac{h}{m \lambda}$

First, convert the mass of each particle from grams to kilograms, and the wavelength $\lambda$ from nanometers to meters.

1. Proton

Mass of proton: $1.673 \times 10^{-24} \, \text{g} = 1.673 \times 10^{-27} \, \text{kg}$

2. Neutron

Mass of neutron: $1.675 \times 10^{-24} \, \text{g} = 1.675 \times 10^{-27} \, \text{kg}$

3. Electron

Mass of electron: $9.109 \times 10^{-28} \, \text{g} = 9.109 \times 10^{-31} \, \text{kg}$

4. Alpha Particle

Mass of alpha particle: $6.645 \times 10^{-24} \, \text{g} = 6.645 \times 10^{-27} \, \text{kg}$

Wavelength $\lambda = 625.0 \, \text{nm} = 625.0 \times 10^{-9} \, \text{m}$

Now let's calculate the speed for each particle. I will compute these values.Here are the calculated speeds for the particles if they had the same wavelength as a photon of orange light ($\lambda = 625.0 \, \text{nm}$):

1. Proton speed: $0.634 \, \text{m/s}$
2. Neutron speed: $0.633 \, \text{m/s}$
3. Electron speed: $1163.86 \, \text{m/s}$
4. Alpha particle speed: $0.160 \, \text{m/s}$

Would you like further details or explanations on any part of this solution?

Here are 5 related questions to deepen your understanding:

1. How does mass affect the speed of a particle with a given wavelength?
2. What is the significance of the de Broglie wavelength in quantum mechanics?
3. Why is the speed of an electron much higher than that of heavier particles like protons and neutrons?
4. How does the speed of light compare to the speeds calculated here for particles?
5. What practical applications use the concept of de Broglie wavelengths?

Tip: The de Broglie wavelength equation shows that as the mass of a particle increases, its speed for a given wavelength decreases.