If travelling at same speeds, which of the following matter waves have the shortest wavelength?(1)electron(2)alphaparticle(He2 )(3)neutron(4)proton?

Question

Answer

Answer

According to the de Broglie's equation :-->
lamda = h/mv
So, higher is the mass shorter will be the wavelength
And , here alpha particle have highest mass , so it has shortest wavelength.

So, correct answer is 'alpha particle (He2+)'...

$$Hope it's help... $$

Answer

Shortest Wavelength of Matter Waves

The wavelength of a matter wave is inversely proportional to its momentum. The de Broglie wavelength of a particle is given by the equation:

λ = h / p

where λ is the wavelength, h is the Planck's constant, and p is the momentum of the particle.

Comparing the Wavelengths of Different Matter Waves

To determine which matter wave has the shortest wavelength when traveling at the same speed, we need to compare the momenta of the given particles.

1. Electron
- The momentum of an electron is given by its mass (m) multiplied by its velocity (v): p = m * v.
- Electrons have a very small mass compared to other particles, such as protons and neutrons.
- Therefore, electrons have a higher momentum for the same velocity, resulting in a shorter wavelength.

2. Alpha Particle (He2)
- An alpha particle consists of two protons and two neutrons, making it heavier than an electron.
- Due to its larger mass, the momentum of an alpha particle is lower than that of an electron for the same velocity.
- Hence, an alpha particle has a longer wavelength compared to an electron.

3. Neutron
- Neutrons have a slightly higher mass than protons, but they are still much heavier than electrons.
- As with the alpha particle, the momentum of a neutron is lower than that of an electron for the same velocity.
- Therefore, a neutron also has a longer wavelength than an electron.

4. Proton
- Protons have the same mass as neutrons but carry a positive charge.
- Similar to electrons and neutrons, the momentum of a proton is lower than that of an electron for the same velocity.
- Consequently, a proton has a longer wavelength compared to an electron.

Conclusion

Among the given particles, the electron has the shortest wavelength when traveling at the same speed. This is because electrons have a very small mass compared to alpha particles, neutrons, and protons. As a result, electrons have a higher momentum and shorter wavelength compared to the other particles.

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