Chapter 5, Problem 5.1P

Interpretation Introduction

Interpretation:

For a nitrogen molecule with mass of $4.65\times {10}^{-23}\text{g}$ moving with a speed of $500\pm 2\text{m}\cdot {\text{s}}^{-1}$, the uncertainty in position has to be calculated.

Concept Introduction:

Heisenberg uncertainty principle states that it impossible to measure the exact position and the momentum of a particle at the same time.  This can be expressed mathematically as shown below.

    $(\Delta x)(\Delta p)\ge \frac{h}{4\pi }$

Where,

    $h$ is the Planck’s constant.

    $\Delta x$ is the uncertainty associated with position.

    $\Delta p$ is the uncertainty associated with momentum.

It is known that the product of mass and velocity is the momentum of the particle.  This can be expressed as shown below.

    $\Delta p=m\Delta v$

Therefore, Heisenberg’s uncertainty principle can also be expressed as shown below.

    $(\Delta x)(m\Delta v)\ge \frac{h}{4\pi }$

Explanation of Solution

Mass of nitrogen $(m)$ molecule is given as $4.65\times {10}^{-23}\text{g}$.  Conversion of this into kilograms gives the value as $4.65\times {10}^{-26}\text{kg}$.

Uncertainty in velocity $(\Delta v)$ is given as $2\text{m}\cdot {\text{s}}^{-1}$.

Uncertainty in position of nitrogen molecule can be calculated as shown below.

    $\begin{array}{c}\Delta x=\frac{h}{4\pi m\Delta v}\\ =\frac{6.626\times {10}^{-34}\text{kg}\cdot {\text{m}}^2\cdot {\text{s}}^{-1}}{4(3.14)(4.65\times {10}^{-26}\text{kg)(}2\text{m}\cdot {\text{s}}^{-1})}\\ =\pm 5.67\times {10}^{-10}\text{m}\end{array}$

Therefore, uncertainty of nitrogen molecule with respect to position is $\pm 5.67\times {10}^{-10}\text{m}$.