Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 5: Problem 18

A magnesium ion, Mg \(^{2+},\) with a charge of \(3.2 \times 10^{-19} \mathrm{Cand}\) an oxide ion, \(\mathrm{O}^{2-},\) with a charge of \(-3.2 \times 10^{-19} \mathrm{C}\) , are separated by a distance of 0.35 \(\mathrm{nm}\) . How much work would be required to increase the separation of the two ions to an infinite distance?

Short Answer

Expert verified
To separate a magnesium ion (Mg\(^{2+}\)) and an oxide ion (O\(^{2-}\)) from a distance of \(0.35 \,\text{nm}\) to an infinite distance, we need to do work equal to the potential energy but with the opposite sign. Using the given charges and the potential energy formula, we calculate the potential energy as \(-3.264 \times 10^{-18} \,\text{J}\). Therefore, the work required to separate the ions is \(3.264 \times 10^{-18}\) Joules.

Step by step solution

01

Understand the problem and identify the given values

We are given the following information: - Charge of magnesium ion (Mg\(^{2+}\)): \(Q_1 = 3.2 \times 10^{-19} \,\text{C}\) - Charge of oxide ion (O\(^{2-}\)): \(Q_2 = -3.2 \times 10^{-19} \,\text{C}\) - Distance between the ions: \(r = 0.35 \,\text{nm}\) (convert it to meters) - The work required to increase the separation of the two ions to an infinite distance.
02

Convert the distance from nm to meters

To use the SI units in our calculation, we need to convert the distance of separation (\(r\)) from nm to meters: $$ r = 0.35 \,\mathrm{nm} \times \frac{1 \times 10^{-9} \,\mathrm{m}}{1 \,\mathrm{nm}} = 3.5 \times 10^{-10} \,\text{m} $$
03

Use the potential energy formula

We will calculate the potential energy \(U\) between the two charged particles (ions) using the following formula: $$ U = k \frac{Q_1 \times Q_2}{r} $$ where \(k\) (the Coulomb constant) is \(8.98755 \times 10^9 \mathrm{Nm^2/C^2}\), \(Q_1\) and \(Q_2\) are the charges, and \(r\) is the separation distance.
04

Calculate the potential energy

Now, substitute the given values into the formula to calculate the potential energy between the two ions: $$ U = (8.98755 \times 10^9 \,\mathrm{Nm^2/C^2}) \times \frac{(3.2 \times 10^{-19}\,\text{C}) \times (-3.2 \times 10^{-19} \,\text{C})}{3.5 \times 10^{-10} \,\text{m}} $$ Calculating the value, we get: $$ U = -3.264 \times 10^{-18} \,\text{J} $$
05

Calculate the work required

To separate the two ions to an infinite distance, we need to do work equal to the potential energy but with the opposite sign (as potential energy at infinite distance is zero). Hence, the work required \(W\) to separate them is: $$ W = -U = 3.264 \times 10^{-18} \,\text{J} $$ Therefore, the work required to increase the separation of the two ions to an infinite distance is \(3.264 \times 10^{-18}\) Joules.

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

Calculate \(\Delta E\) and determine whether the process is endothermic or exothermic for the following cases: (a) \(q=0.763 \mathrm{kJ}\) and \(w=-840 \mathrm{J} .\) (b) A system releases 66.1 \(\mathrm{kJ}\) of heat to its surroundings while the surroundings do 44.0 \(\mathrm{kJ}\) of work on the system.

(a) What is meant by the term standard conditions with reference to enthalpy changes? (b) What is meant by the term enthalpy of formation? (c) What is meant by the term standard enthalpy of formation?

(a) According to the first law of thermodynamics, what quantity is conserved? (b) What is meant by the internal energy of a system? (c) By what means can the internal energy of a closed system increase?

A 2.200 -g sample of quinone \(\left(\mathrm{C}_{6} \mathrm{H}_{4} \mathrm{O}_{2}\right)\) is burned in a bomb calorimeter whose total heat capacity is 7.854 \(\mathrm{kJ} / \mathrm{c}\) . The temperature of the calorimeter increases from 23.44 to \(30.57^{\circ} \mathrm{C}\) . What is the heat of combustion per gram of quinone? Per mole of quinone?

(a) Which releases the most energy when metabolized, 1 \(\mathrm{g}\) of carbohydrates or 1 \(\mathrm{g}\) of fat? (b) A particular chip snack food is composed of 12\(\%\) protein, 14\(\%\) fat, and the rest carbohydrate. What percentage of the calorie content of this food is fat? (c) How many grams of protein provide the same fuel value as 25 of fat?
See all solutions

Recommended explanations on Chemistry Textbooks

Physical Chemistry

Read Explanation

Chemistry Branches

Read Explanation

Making Measurements

Read Explanation

Organic Chemistry

Read Explanation

Chemical Reactions

Read Explanation

Chemical Analysis

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