Chapter 11, Problem 100AP

Interpretation Introduction

(a)

Interpretation:

The de-Broglie wavelength is to be calculated.

Concept Introduction:

In $1924$, de-Broglie suggested the dual nature of microscopic particles and compared the energy value of both particle and wave nature particles. According to him, the wavelength of an electron $\left(\text{λ}\right)$ moving with a speed of v is:

$\text{λ}=\frac{\text{h}}{\text{m}\text{v}}$

Where,

• $\text{h}$ is Plank’s constant.
• $\text{m}$ is the mass of electron.

Answer to Problem 100AP

The de-Broglie wavelength is calculated as $\text{2}\text{.7}\times {\text{10}}^{-\text{12}}{\text{ m}}^{-1}$.

Explanation of Solution

Mass of electron $=9.10\times {\text{10}}^{-31}\text{kg}$

Speed of light $={\text{3×10}}^{\text{8}}{\text{ms}}^{-\text{1}}$

Given: The speed of electron is $0.9$ times the speed of light.

So the speed of electron, $\text{v}=0.9\times 3\times {10}^8{\text{ms}}^{\text{-1}}$

The formula for the calculation of de-Broglie wavelength is:

$\text{λ}=\frac{\text{h}}{\text{m}\text{v}}$

The value of Plank’s constant, $\text{h}=6.63\times {10}^{-34}\text{J}\cdot \text{s}$

Applying the formula for calculating wavelength:

$\begin{array}{l}\text{λ}=\frac{\text{h}}{\text{m}\text{v}}\\ \text{λ}=\frac{6.63\times {10}^{-34}\text{J}\cdot \text{s}}{\left(9.10\times {10}^{-31}\text{kg}\right)\left(0.9\times 3\times {10}^8{\text{ms}}^{-1}\right)}\\ \text{λ}=\text{2}\text{.7}\times {\text{10}}^{-\text{12}}{\text{ m}}^{-1}\end{array}$.

Interpretation Introduction

(b)

Interpretation:

The de-Broglie wavelength is to be calculated and the reason as to why macroscopic particles cannot be treated as wave particle is to be stated.

Concept Introduction:

In $1924$, de-Broglie suggested the dual nature of microscopic particles and compared the energy value of both particle and wave nature particles. According to him, the wavelength of an electron $\left(\text{λ}\right)$ moving with a speed of v is:

$\text{λ}=\frac{\text{h}}{\text{m}\text{v}}$

Where,

• $\text{h}$ is Plank’s constant.
• $\text{m}$ is the mass of electron.

Answer to Problem 100AP

The de-Broglie wavelength is calculated as $4.42\times {\text{10}}^{-34}{\text{m}}^{-1}$ and such magnitude is practically impossible to achieve.

Explanation of Solution

Given:

Mass of ball $=150\text{g}$

Conversion of grams to kilograms can be done as:

$1000\text{g}=1\text{kg}$

So,

$150\text{g}\text{=}\frac{150}{1000}\text{kg}\text{=}\text{0}\text{.15}\text{kg}$

Speed of ball $=\text{10}{\text{ms}}^{-\text{1}}$

The formula for the calculation of de-Broglie wavelength is:

$\text{λ}=\frac{\text{h}}{\text{m}\text{v}}$

The value of Plank’s constant, $\text{h}=6.63\times {10}^{-34}\text{J}\cdot \text{s}$

Applying the formula for calculating wavelength:

$\begin{array}{l}\text{λ}=\frac{\text{h}}{\text{m}\text{v}}\\ \text{λ}=\frac{6.63\times {10}^{-34}\text{J}\cdot \text{s}}{\left(\text{0}\text{.150}\text{kg}\right)\left(10{\text{ms}}^{-1}\right)}\\ \text{λ}=4.42\times {\text{10}}^{-34}{\text{m}}^{-1}\end{array}$

For a ball of mass $0.150\text{kg}$, it is practically impossible to have a wavelength of such magnitude and therefore, the wave theory of particles cannot be applied to the macroscopic particles.

Interpretation Introduction

(b)

Interpretation:

The de-Broglie wavelength is to be calculated and the reason as to why macroscopic particles cannot be treated as wave particle is to be stated.

Concept Introduction:

In $1924$, de-Broglie suggested the dual nature of microscopic particles and compared the energy value of both particle and wave nature particles. According to him, the wavelength of an electron $\left(\text{λ}\right)$ moving with a speed of v is:

$\text{λ}=\frac{\text{h}}{\text{m}\text{v}}$

Where,

• $\text{h}$ is Plank’s constant.
• $\text{m}$ is the mass of electron.

Answer to Problem 100AP

The de-Broglie wavelength is calculated as $4.42\times {\text{10}}^{-34}{\text{m}}^{-1}$ and such magnitude is practically impossible to achieve.

Explanation of Solution

Given:

Mass of ball $=150\text{g}$

Conversion of grams to kilograms can be done as:

$1000\text{g}=1\text{kg}$

So,

$150\text{g}\text{=}\frac{150}{1000}\text{kg}\text{=}\text{0}\text{.15}\text{kg}$

Speed of ball $=\text{10}{\text{ms}}^{-\text{1}}$

The formula for the calculation of de-Broglie wavelength is:

$\text{λ}=\frac{\text{h}}{\text{m}\text{v}}$

The value of Plank’s constant, $\text{h}=6.63\times {10}^{-34}\text{J}\cdot \text{s}$

Applying the formula for calculating wavelength:

$\begin{array}{l}\text{λ}=\frac{\text{h}}{\text{m}\text{v}}\\ \text{λ}=\frac{6.63\times {10}^{-34}\text{J}\cdot \text{s}}{\left(\text{0}\text{.150}\text{kg}\right)\left(10{\text{ms}}^{-1}\right)}\\ \text{λ}=4.42\times {\text{10}}^{-34}{\text{m}}^{-1}\end{array}$

For a ball of mass $0.150\text{kg}$, it is practically impossible to have a wavelength of such magnitude and therefore, the wave theory of particles cannot be applied to the macroscopic particles.

Interpretation Introduction

(c)

Interpretation:

The de-Broglie wavelength is to be calculated and the reason as to why macroscopic particles cannot be treated as wave particle is to be stated.

Concept Introduction:

In $1924$, de-Broglie suggested the dual nature of microscopic particles and compared the energy value of both particle and wave nature particles. According to him, the wavelength of an electron $\left(\text{λ}\right)$ moving with a speed of v is:

$\text{λ}=\frac{\text{h}}{\text{m}\text{v}}$

Where,

• $\text{h}$ is Plank’s constant.
• $\text{m}$ is the mass of electron.

Answer to Problem 100AP

The de-Broglie wavelength is calculated to be as $1.59\times {\text{10}}^{-35}{\text{m}}^{-1}$ and such magnitude is practically impossible to achieve.

Explanation of Solution

Given:

Mass of person $=75\text{kg}$

Speed of person $=2{\text{kmh}}^{-\text{1}}$

Conversion of speed in kilometer per hour to speed in meter per second can be done as:

$\begin{array}{l}1\text{km}=1000\text{m}\\ \text{1}\text{hour}=3600\mathrm{s}\end{array}$

So,

$2{\text{kmh}}^{-1}\text{=}\frac{2\times 1000\text{m}}{3600\text{s}}\text{=}\frac{10}{18}{\text{ms}}^{-1}$

The formula for the calculation of de-Broglie wavelength is:

$\text{λ}=\frac{\text{h}}{\text{m}\text{v}}$

The value of Plank’s constant, $\text{h}=6.63\times {10}^{-34}\text{J}\cdot \text{s}$

Applying the formula for calculating wavelength:

$\begin{array}{l}\text{λ}=\frac{\text{h}}{\text{m}\text{v}}\\ \text{λ}=\frac{6.63\times {10}^{-34}\text{Js}}{\left(75\text{kg}\right)\left(\frac{10}{18}{\text{ms}}^{-1}\right)}\\ \text{λ}=1.59\times {\text{10}}^{-35}{\text{m}}^{-1}\end{array}$

For a person of mass $75\text{kg}$, it is practically impossible to have a wavelength of such magnitude and therefore, the wave theory of particles cannot be applied to the macroscopic particles.