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Chapter 19: Problem 4

In an ionic solution, a current consists of \(\mathrm{Ca}^{2}+\) ions (of charge \(+2 e\) ) and \(\mathrm{Cl}^{-}\) ions (of charge \(-e\) ) traveling in opposite directions. If \(5.11 \times 10^{18} \mathrm{Cl}^{-}\) ions go from \(A\) to \(B\) every \(0.50 \mathrm{~min},\) while \(3.24 \times 10^{18} \mathrm{Ca}^{2+}\) ions move from \(B\) to \(A,\) what is the current (in mA) through this solution, and in which direction (from \(A\) to \(B\) or from \(B\) to \(A\) ) is it going?

Short Answer

Expert verified
Current is 25.17 mA from B to A.

Step by step solution

01

Understand the Problem

We have two types of ions in an ionic solution with given ion counts and movement directions over a specified time. The task is to determine the current magnitude and direction.
02

Identify Charge and Time Information

The charge of one electron is \( e = 1.6 \times 10^{-19} \text{ C} \). We have \( 5.11 \times 10^{18} \text{ Cl}^{-} \) ions moving from A to B each 0.50 min and \( 3.24 \times 10^{18} \text{ Ca}^{2+} \) ions moving from B to A, for the same 0.50 min, which is \(30 \text{ s}\).
03

Calculate Total Charge Transfer for Each Ion Type

The total charge carried by \(\text{Cl}^{-}\) ions is \(-5.11 \times 10^{18} \times -1.6 \times 10^{-19} \text{ C} = 5.11 \times 1.6 \times 10^{-1} \text{ C}\). The total charge carried by \(\text{Ca}^{2+}\) ions is \(3.24 \times 10^{18} \times 2 \times 1.6 \times 10^{-19} \text{ C} = 3.24 \times 3.2 \times 10^{-1} \text{ C}\).
04

Calculate Net Charge Transfer and Determine Current Direction

Net charge transfer is the difference between the charges: \( 5.11 \times 1.6 \times 10^{-1} - 3.24 \times 3.2 \times 10^{-1} \). It results in \(0.2816 - 1.0368 = -0.7552\) C. Since it's negative, current is from B to A.
05

Compute Current in Milliamps

Convert charge to current using \( I = \frac{Q}{t} \), with \( Q = -0.7552 \text{ C} \) and \( t = 30 \text{ s} \): \( I = \frac{-0.7552}{30} \approx -0.02517 \text{ A} \) or \(-25.17 \text{ mA}\). The negative sign confirms direction from B to A.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Ionic Solution
An ionic solution is a liquid that contains free-moving charged particles. These charged particles, known as ions, are the reason ionic solutions can conduct electricity. Here鈥檚 how it works: when you dissolve a salt, like calcium chloride, in water, it splits into ions.
  • This separation allows ions to move freely.
  • Free-moving ions create an electrical pathway.
  • Their movement under the influence of an electric field results in an ionic current.
Particles in an ionic solution with different charges will move toward opposite charges when an electric field is applied. This is why you often see ions moving in different directions simultaneously.
Calcium Ions
Calcium ions, denoted as \(Ca^{2+}\), are ions with a charge of \(+2e\). In an ionic solution, calcium ions play a crucial role in conducting electricity. Their positive charge means they are attracted to areas of negative charge.
  • Calcium ions move opposite to negatively charged ions.
  • They contribute to the positive side of the current in an ionic solution.
  • Calcium ions are created when calcium chloride is dissolved, separating into calcium and chloride ions.
Understanding the movement of calcium ions helps explain electricity flow direction in the solution.
Chloride Ions
Chloride ions, represented by \(Cl^{-}\), have a negative charge, carrying \(-e\). These ions are important in forming the ionic current when placed in the ionic solutions.
  • They move towards positively charged areas when an electromagnetic field is applied.
  • Their motion is opposite that of calcium ions.
  • In our exercise, they contribute to the moving current from A to B because of their negative charge being attracted to positive charges.
Chloride ions provide the balancing negative charge in ionic solutions, counteracting the positively charged ions like calcium, and thus creating an overall ionic current.
Current Direction
The direction of current in an ionic solution is determined by the net movement of ions and their charges. When ions move, they carry charges, thereby creating a current.
  • If more positive charges move in one direction than negative, the current follows the positive charge flow.
  • Conversely, if more negative charges move, current flows in the same direction.
In the example given, the calculation showed that there is a net negative charge, meaning more charge flowed from A to B due to chloride ions than that returned by calcium ions. Hence, the current in our solution flows from B to A, opposing initial assumptions, guided by the majority charge movement.

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Most popular questions from this chapter

An automobile starter motor is connected to a \(12.0 \mathrm{~V}\) battery. When the starter is activated it draws 150 A of current, and the battery voltage drops to \(7.0 \mathrm{~V}\). What is the battery's internal resistance?

Some types of spiders build webs that consist of threads made of dry silk coated with a solution of a variety of compounds. This coating leaves the threads, which are used to capture prey, hygroscopic-that is, they attract water from the atmosphere. It has been hypothesized that this aqueous coating makes the threads good electrical conductors. To test the electrical properties of coated thread, researchers placed a \(5 \mathrm{~mm}\) length of thread between two electrical contacts. The researchers stretched the thread in \(1 \mathrm{~mm}\) increments to more than twice its original length, and then allowed it to return to its original length, again in \(1 \mathrm{~mm}\) increments. Some of the resistance measurements are given in the table: $$\begin{array}{l|cccccccc}\begin{array}{l}\text { Resistance of } \\\\\text { thread }\left(10^{9} \Omega\right)\end{array} & 9 & 19 & 41 & 63 & 102 & 76 & 50 & 24 \\ \hline \begin{array}{l}\text { Length of } \\\\\text { thread }(\mathrm{mm})\end{array} & 5 & 7 & 9 & 11 & 13 & 9 & 7 & 5\end{array}$$ What is the best explanation for the behavior exhibited in the data? A. Longer threads can carry more current than shorter threads and so make better electrical conductors. B. The thread stops being a conductor when it is stretched to 13 \(\mathrm{mm}\) due to breaks that occur in the thin coating. C. As the thread is stretched, the coating thins and its resistance increases; as the thread is relaxed, the coating returns nearly to its original state. D. The resistance of the thread increases with distance from the end of the thread.

A \(500.0 \Omega\) resistor is connected in series with a capacitor. What must be the capacitance of the capacitor to produce a time constant of \(2.00 \mathrm{~s} ?\)

A refrigerator draws 3.5 A of current while operating on a \(120 \mathrm{~V}\) power line. If the refrigerator runs \(50 \%\) of the time and electric power costs \(\$ 0.12\) per \(\mathrm{kWh}\), how much does it cost to run this refrigerator for a 30 -day month?

This procedure is not recommended! You'll see why after you work the problem. You are on an aluminum ladder that is standing on the ground, trying to fix an electrical connection with a metal screwdriver that has a metal handle. Your body is wet because you are sweating from the exertion; therefore, it has a resistance of \(1.0 \mathrm{k} \Omega .\) (a) If you accidentally touch the "hot" wire connected to the \(120 \mathrm{~V}\) line, how much current will pass through your body? Is this amount enough to be dangerous? (The maximum safe current is about \(5 \mathrm{~mA} .\) ) (b) How much electric power is delivered to your body?
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