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Chapter 8: Problem 126

If traveling at equal speeds, which of the following matter waves has the longest wavelength? Explain. (a) electron; (b) proton; (c) neutron; (d) \(\alpha\) particle \(\left(\mathrm{He}^{2+}\right)\).

Short Answer

Expert verified
The electron (a) has the longest wavelength when traveling at equal speeds since it has the smallest mass.

Step by step solution

01

Identify the Masses of the Particles

Firstly, determine the masses of the particles in question. The respective masses for the electron (e), proton(p), neutron(n) and alpha (α) particle are: (e) \( 9.11E-31 \) kg, (p) \( 1.67E-27 \) kg, (n) \( 1.67E-27 \) kg, (α) \( 6.64E-27 \) kg.
02

Apply the de Broglie equation

Since all the particles are traveling at equal speeds, according to the de Broglie equation \( \lambda = \frac{h}{mv} \), the particle with the least mass will have the longest wavelength.
03

Determine the Particle with the Smallest Mass

Comparing the masses of the particles, it is observed that the electron has the smallest mass.
04

Conclude

Therefore, the electron (option a) having the smallest mass, traveling at equal speeds as the rest, will have the longest wavelength according to the de Broglie equation.

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