Name: Nuclear Chain Reaction
Uploaded: 2019-07-19
Duration: 8 min 15 s
Description: Have you ever made a row of dominoes fall by pushing the first one? It is like a chain reaction. Learn about another type of chain reaction that happens on t...

Question

Calculate the de Broglie wavelengths of an electron, proton and uranium atom, all having the same kinetic energy

100 eV .

Mass of the electron is $m = 9.1 \times 10^{- 31} kg$ , mass of proton is $1.67 \times 10^{- 27} kg$ , mass of uranium atom is $3.95 \times 10^{- 25} kg$ , Planck's constant $h = 6.626 \times 10^{- 34} J {Hz}^{- 1}$ .

Accepted Answer

From the question, the kinetic energy $T = 100 eV = 100 \times 1.6 \times 10^{- 19} J$ .

Now, we know that kinetic energy of electron is,

$T = \frac{1}{2} m_{e} v_{e}^{2}$ , where symbols have their usual meaning.

Now, the relation between kinetic energy and momentum of electron is,

$p_{e} = m_{e} v_{e} \Rightarrow v_{e} = \frac{p_{e}}{m_{e}} T = \frac{1}{2} \times \frac{p_{e}^{2}}{m_{e}}$

Now, same as for proton and uranium, the formula for proton is,

$T = \frac{1}{2} \times \frac{p_{p}^{2}}{m_{p}}$ and that for uranium is, $T = \frac{1}{2} \times \frac{p_{ur}^{2}}{m_{ur}}$

For electron,

$λ_{e} = \frac{h}{p_{e}} = \frac{h}{\sqrt{2 m_{e} T}} = \frac{6.67 \times 10^{- 34}}{\sqrt{2 \times 9.1 \times 10^{- 31} \times 100 \times 1.6 \times 10^{- 19}}} = 122.6 \times 10^{- 12} m = 122.6 pm$

For proton,

$λ_{p} = \frac{h}{p_{p}} = \frac{h}{\sqrt{2 m_{p} T}} = \frac{6.67 \times 10^{- 34}}{\sqrt{2 \times 1.67 \times 10^{- 27} \times 100 \times 1.6 \times 10^{- 19}}} = 2.86 \times 10^{- 12} m = 2.86 pm$

For uranium,

$λ_{ur} = \frac{h}{p_{ur}} = \frac{h}{\sqrt{2 m_{ur} T}} = \frac{6.67 \times 10^{- 34}}{\sqrt{2 \times 238 \times 1.67 \times 10^{- 27} \times 100 \times 1.6 \times 10^{- 19}}} = 0.185 \times 10^{- 12} m = 0.185 pm$

using $T = 1.602 \times 10^{- 17} J$ (where we have used a mass number of $238$ for the $U$ nucleus).

I E Irodov Solutions for Chapter: ATOMIC AND NUCLEAR PHYSICS, Exercise 2: WAVE PROPERTIES OF PARTICLES. SCHRODINGER EQUATION

I E Irodov Physics Solutions for Exercise - I E Irodov Solutions for Chapter: ATOMIC AND NUCLEAR PHYSICS, Exercise 2: WAVE PROPERTIES OF PARTICLES. SCHRODINGER EQUATION

Questions from I E Irodov Solutions for Chapter: ATOMIC AND NUCLEAR PHYSICS, Exercise 2: WAVE PROPERTIES OF PARTICLES. SCHRODINGER EQUATION with Hints & Solutions

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