Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 6: Problem 51

Using Heisenberg's uncertainty principle, calculate the uncertainty in the position of (a) a 1.50 -mg mosquito moving at a speed of $1.40 \mathrm{~m} / \mathrm{s}\( if the speed is known to within \)\pm 0.01 \mathrm{~m} / \mathrm{s} ;\( (b) a proton moving at a speed of \)(5.00 \pm 0.01) \times 10^{4} \mathrm{~m} / \mathrm{s}$ (The mass of a proton is given in the table of fundamental constants in the inside cover of the text.)

Short Answer

Expert verified
The uncertainties in position for both the mosquito and the proton are: a) \(\Delta x_{\text{mosquito}}\) is at least approximately \(3.5 \times 10^{-26}\, \text{m}\) b) \(\Delta x_{\text{proton}}\) is at least approximately \(3.2 \times 10^{-13}\, \text{m}\)

Step by step solution

01

Data

Mass of mosquito: \(m = 1.50\,\text{mg} = 1.50 \times 10^{-6} \,\text{kg}\) Uncertainty in speed: \(\Delta v = 0.01\,\text{m/s}\)
02

Calculate uncertainty in momentum (mosquito)

Using the formula \(\Delta p = m\Delta v\), we get: \[\Delta p_{\text{mosquito}} = (1.50 \times 10^{-6}\,\text{kg})(0.01\,\text{m/s}) = 1.50 \times 10^{-8}\,\text{kg m/s}\] b) Proton
03

Data

Mass of proton: \(m_{\text{proton}}= 1.67 \times 10^{-27} \,\text{kg}\) Uncertainty in speed: \(\Delta v_{\text{proton}} = 0.01 \times 10^4\,\text{m/s}\)
04

Calculate uncertainty in momentum (proton)

Using the formula \(\Delta p = m\Delta v\), we get: \[\Delta p_{\text{proton}} = (1.67 \times 10^{-27}\,\text{kg})(0.01 \times 10^4\,\text{m/s}) = 1.67 \times 10^{-22}\,\text{kg m/s}\] ##Step 2: Calculate the uncertainty in position## We will now use Heisenberg's uncertainty principle formula \(\Delta x \Delta p \geq \frac{\hbar}{2}\) to find the uncertainties in position.
05

Heisenberg's uncertainty principle formula

Using the uncertainty principle formula and solving for \(\Delta x\), we get: \[\Delta x \geq \frac{\hbar}{2\Delta p}\] a) Mosquito
06

Calculate uncertainty in position (mosquito)

For the mosquito, we have \(\Delta p_{\text{mosquito}} = 1.50 \times 10^{-8}\,\text{kg m/s}\). Using the inequality above, we get: \[\Delta x_{\text{mosquito}} \geq \frac{1.054 \times 10^{-34}\, \text{J s}}{2(1.50 \times 10^{-8}\,\text{kg m/s})} \approx 3.5 \times 10^{-26}\, \text{m}\] b) Proton
07

Calculate uncertainty in position (proton)

For the proton, we have \(\Delta p_{\text{proton}} = 1.67 \times 10^{-22}\,\text{kg m/s}\). Using the inequality above, we get: \[\Delta x_{\text{proton}} \geq \frac{1.054 \times 10^{-34}\, \text{J s}}{2(1.67 \times 10^{-22}\,\text{kg m/s})} \approx 3.2 \times 10^{-13}\, \text{m}\]
08

Final answers

The uncertainties in position for both the mosquito and the proton are: a) \(\Delta x_{\text{mosquito}}\) is at least approximately \(3.5 \times 10^{-26}\, \text{m}\) b) \(\Delta x_{\text{proton}}\) is at least approximately \(3.2 \times 10^{-13}\, \text{m}\)

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

In January 2006, the New Horizons space probe was launched from Earth with the mission to perform a flyby study of Pluto. The arrival at the dwarf planet was estimated to happen after nine years, in 2015 . The distance between Earth and Pluto varies depending on the location of the planets in their orbits, but at their closest, the distance is 4.2 billion kilometers \((2.6\) billion miles). Calculate the minimum amount of time it takes for a transmitted signal from Pluto to reach the Earth.

For orbitals that are symmetric but not spherical, the contour representations (as in Figures 6.23 and 6.24 ) suggest where nodal planes exist (that is, where the electron density is zero). For example, the \(p_{x}\) orbital has a node wherever \(x=0\). This equation is satisfied by all points on the \(y z\) plane, so this plane is called a nodal plane of the \(p_{x}\) orbital. (a) Determine the nodal plane of the \(p_{z}\) orbital. (b) What are the two nodal planes of the \(d_{x y}\) orbital? (c) What are the two nodal planes of the \(d_{x^{2}-y^{2}}\) orbital?

Determine which of the following statements are false and correct them. (a) The frequency of radiation increases as the wavelength increases. (b) Electromagnetic radiation travels through a vacuum at a constant speed, regardless of wavelength. (c) Infrared light has higher frequencies than visible light. (d) The glow from a fireplace, the energy within a microwave oven, and a foghorn blast are all forms of electromagnetic radiation.

Give the values for \(n, l,\) and \(m_{l}\) for \((\mathbf{a})\) each orbital in the \(3 p\) subshell, (b) each orbital in the \(4 f\) subshell.

Sketch the shape and orientation of the following types of orbitals: \((\mathbf{a}) s,(\mathbf{b}) p_{z},(\mathbf{c}) d_{x y}\)
See all solutions

Recommended explanations on Chemistry Textbooks

Chemical Reactions

Read Explanation

Ionic and Molecular Compounds

Read Explanation

Nuclear Chemistry

Read Explanation

Organic Chemistry

Read Explanation

Chemical Analysis

Read Explanation

Inorganic Chemistry

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.

Sign-up for free