Chapter 1, Problem 1B.28E

(a)

Interpretation Introduction

Interpretation:

The minimum uncertainty in the speed of an electron confined in a nanoparticle with a diameter of $2.00\times {10}^{2}nm$ has to be calculated.

Concept Introduction:

Uncertainty principle:

Uncertainty principle states that if the location of a particle is known to within an uncertainty $\Delta x$, then the linear momentum, p, parallel to the x – axis can be known simultaneously only to within an uncertainty $\Delta p.$

Mathematically, $\Delta p\times \Delta x\ge \frac{1}{2}\hslash$

Where, $\Delta x$ is the uncertainty in position and $\Delta p$ is the uncertainty in momentum.

Answer to Problem 1B.28E

The minimum uncertainty in the speed of an electron confined within a lead atom of diameter $350pm$ is $1.65\times 10^{5}m/s.$

Explanation of Solution

Given that, the uncertainty in position of an electron is $2.00\times {10}^{2}nm.$  The mass of electron is $9.109\times {10}^{-31}kg.$

The uncertainty principle says that,

  $\begin{array}{l}\Delta p\times \Delta x=\frac{1}{2}\hslash \\ \\ m\Delta v\times \Delta x=\frac{1}{2}\hslash \\ \\ \Delta v=\frac{\hslash }{2m\Delta x}\end{array}$

By plugging all data in the above equation, the value of uncertainty in the speed of electron can be calculated.

  $\begin{array}{l}\Delta v=\frac{\hslash }{2m\Delta x}=\frac{1.05457\times {10}^{-34}J.s}{2\times \left(9.109\times {10}^{-31}kg\right)\times \left(2.00\times {10}^{-7}m\right)}\\ \\ \Delta v=289.43\times \frac{J.s}{kg.m}\\ \\ \Delta v=289.43\times \frac{kg.m^2.s^{-2}.s}{kg.m}=289.43m/s\\ \\ \Delta v=289.43m/s\end{array}$

Therefore, the minimum uncertainty in the speed of an electron confined within a nanoparticle of diameter $2.00\times {10}^{2}nm$ is $289.43m/s.$

(b)

Interpretation Introduction

Interpretation:

The minimum uncertainty in the speed of a $L{i}^{+}$ confined in a nanoparticle with a diameter of $2.00\times {10}^{2}nm$ has to be calculated.

Concept Introduction:

Refer to part (a).

Answer to Problem 1B.28E

The minimum uncertainty in the speed of a confined within a $L{i}^{+}$ nanoparticle of diameter $2.00\times {10}^{2}nm$ is $2.25\times 10^{-5}m/s.$

Explanation of Solution

Given that, the uncertainty in position of a $L{i}^{+}$ is $2.00\times {10}^{2}nm.$  The mass of a $L{i}^{+}$ ion can be calculated as given below.

$L{i}^{+}$ has two electrons, three protons and four neutrons.

  $\begin{array}{l}massofL{i}^{+}=2\times massofelectron+3\times massofproton+4\times \times massofneutron\\ \\ massofL{i}^{+}=2\times 0.000548579909u+3\times 1.00727647u+4\times 1.00866491588u\\ \\ massofL{i}^{+}=7.06u\\ \\ 1molL{i}^{+}=6.023\times {10}^{23}ions\\ \\ massof1L{i}^{+}ion=7.06u\times \left(\frac{1molL{i}^{+}}{6.023\times {10}^{23}L{i}^{+}ions}\right)\\ \\ massof1L{i}^{+}ion=1.17\times 10^{-23}kg\end{array}$

The uncertainty principle says that,

  $\begin{array}{l}\Delta p\times \Delta x=\frac{1}{2}\hslash \\ \\ m\Delta v\times \Delta x=\frac{1}{2}\hslash \\ \\ \Delta v=\frac{\hslash }{2m\Delta x}\end{array}$

By plugging all data in the above equation, the value of uncertainty in the speed of a $L{i}^{+}$ can be calculated.

  $\begin{array}{l}\Delta v=\frac{\hslash }{2m\Delta x}=\frac{1.05457\times {10}^{-34}J.s}{2\times \left(\text{1}{\text{.17×10}}^{\text{-23}}kg\right)\times \left(2.00\times {10}^{-7}m\right)}\\ \\ \Delta v=2.25\times {10}^{-5}\frac{J.s}{kg.m}\\ \\ \Delta v=2.25\times {10}^{-5}\times \frac{kg.m^2.s^{-2}.s}{kg.m}=2.25\times {10}^{-5}m/s\\ \\ \Delta v=2.25\times 10^{-5}m/s\end{array}$

Therefore, the minimum uncertainty in the speed of a confined within a $L{i}^{+}$ nanoparticle of diameter $2.00\times {10}^{2}nm$ is $2.25\times 10^{-5}m/s.$

(c)

Interpretation Introduction

Interpretation:

The one which could be specified more accurately in a nanoparticle: the speed of an electron or the speed of a $L{i}^{+}$ ion has to be determined.

Concept Introduction:

Refer to part (a).

Explanation of Solution

The uncertainty in speed of an electron and a $L{i}^{+}$ ion confined in a nanoparticle with a diameter of same size are $\text{289}\text{.43}\text{m/s}$ and $\text{2}{\text{.25×10}}^{\text{-5}}\text{m/s}$ respectively.

The uncertainty in speed of a $L{i}^{+}$ ion is very small in comparison to an electron confined in a nanoparticle of same size.  Therefore, the speed of an electron could be specified more accurately in a nanoparticle.