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In an ionic solution, a current consists of Ca\(^2+\) ions (of charge \(+2e\)) and Cl\(^-\) ions (of charge \(-e\)) traveling in opposite directions. If \(5.11 \times 1018\) Cl\(^-\) ions go from \(A\) to \(B\) every 0.50 min, while 3.24 \(\times\) 10\(^{18}\) 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
The current is approximately 0.752 mA, directed from A to B.

Step by step solution

01

Calculate Charge of Cl- ions

The charge on one Cl\(^-\) ion is \(-e\), where \(e\) is the elementary charge \(1.6 \times 10^{-19} \text{ C}\). The total charge from Cl\(^-\) ions is \(-5.11 \times 10^{18} \times 1.6 \times 10^{-19} \text{ C}\).
02

Calculate Charge of Ca2+ ions

The charge on one Ca\(^{2+}\) ion is \(+2e\). Therefore, the total charge from Ca\(^{2+}\) ions is \(3.24 \times 10^{18} \times 2 \times 1.6 \times 10^{-19} \text{ C}\).
03

Calculate Total Charge Exchange

Since Cl\(^-\) ions and Ca\(^{2+}\) ions move in opposite directions, their charges will add up when calculating the current: \(Q_{\text{total}} = -5.11 \times 10^{18} \times 1.6 \times 10^{-19} + 3.24 \times 10^{18} \times 2 \times 1.6 \times 10^{-19}\).
04

Convert Time Units

Convert 0.50 minutes to seconds: \(0.50 \times 60 = 30 \text{ seconds}\).
05

Calculate Current

The current \(I\) is found by dividing the total charge by the time in seconds: \(I = \frac{Q_{\text{total}}}{30} \text{ A}\).
06

Determine Current Direction

Since the overall charge is positive, the current direction is from \(A\) to \(B\).

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Key Concepts

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

Ionic Charge Calculation
In the context of ionic solutions, understanding the calculation of ionic charges is crucial. Let's break it down:
  • Individual Charge: Each ion in a solution has a specific charge based on the type of ion. In our example, the Cl\(^-\) ion possesses a charge of \(-e\), where \(e\) represents the elementary charge, or \(1.6 \times 10^{-19} \text{ C}\). The Ca\(^{2+}\) ion holds a charge of \(+2e\).
  • Total Charge for Multiple Ions: To determine the total charge contributed by an entire set of ions, multiply the number of these ions by the charge of a single ion. For instance, 5.11 \(\times 10^{18}\) Cl\(^-\) ions collectively have a charge of \(-5.11 \times 10^{18} \times 1.6 \times 10^{-19} \text{ C}\).
  • Combining Charges: When calculating total charge exchange, consider the direction of ion movement. Opposite charges and directions result in the summation of charges, reflecting the total charge flow in the solution.
Understanding these basics allows you to calculate how much charge ions contribute to a current, an essential step in evaluating the current through a solution.
Current Direction
The direction of current in an ionic solution is determined by calculating the net charge movement resulting from all participating ions. It's typically seen that:
  • The charge carriers (ions) lose or gain electrons, leading to a movement which constitutes the electric current.
  • In solutions with ions moving in opposite directions, the overall current direction is aligned with the net positive charge movement.
  • For the given problem:
    • Cl\(^-\) ions move from \(A\) to \(B\), carrying negative charge;
    • Ca\(^{2+}\) ions move from \(B\) to \(A\), carrying positive charge.
  • After computing the resultant charges, if the total charge is positive, as in our example, the current flows from \(A\) to \(B\).
Recognizing current direction is critical as it influences how circuits and systems are designed and analyzed.
Time Conversion in Physics
Time conversion is a key skill in physics, especially when dealing with rates of change like current, which is defined as charge per unit time. Here's a breakdown:
  • Units and Relevance: Time in physics is often measured in seconds because it is part of the International System of Units (SI). Most formulas, particularly those involving electrical current, utilize seconds.
  • Converting Units: When time is given in units like minutes or hours, it needs conversion to seconds to align with SI units. The conversion rate from minutes to seconds, for instance, is \(1 \text{ minute} = 60 \text{ seconds}\).
  • Application: In our case, 0.50 minutes must be converted to 30 seconds. This step is vital to correctly calculate the current (in Amperes), using the formula \(I = \frac{Q_{\text{total}}}{t}\), where \(t\) is in seconds.
By mastering time conversions, you'll ensure accurate calculations in physics, leading to precise results and analyses.

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Most popular questions from this chapter

An overhead transmission cable for electrical power is 2000 m long and consists of two parallel copper wires, each encased in insulating material. A short circuit has developed somewhere along the length of the cable where the insulation has worn thin and the two wires are in contact. As a power-company employee, you must locate the short so that repair crews can be sent to that location. Both ends of the cable have been disconnected from the power grid. At one end of the cable (point \(A\)), you connect the ends of the two wires to a 9.00-V battery that has negligible internal resistance and measure that 2.86 A of current flows through the battery. At the other end of the cable (point \(B\)), you attach those two wires to the battery and measure that 1.65 A of current flows through the battery. How far is the short from point A?

Nerve cells transmit electric signals through their long tubular axons. These signals propagate due to a sudden rush of Na\({^+}\) ions, each with charge \(+e\), into the axon. Measurements have revealed that typically about 5.6 \(\times\) 10\(^{11}\) Na\({^+}\) ions enter each meter of the axon during a time of 10 ms. What is the current during this inflow of charge in a meter of axon?

A copper wire has a square cross section 2.3 mm on a side. The wire is 4.0 m long and carries a current of 3.6 A. The density of free electrons is 8.5 \(\times\) 10\(^{28}\)/m\({^3}\). Find the magnitudes of (a) the current density in the wire and (b) the electric field in the wire. (c) How much time is required for an electron to travel the length of the wire?

An electrical conductor designed to carry large currents has a circular cross section 2.50 mm in diameter and is 14.0 m long. The resistance between its ends is 0.104\(\Omega\). (a) What is the resistivity of the material? (b) If the electric-field magnitude in the conductor is 1.28 V/m, what is the total current? (c) If the material has \(8.5 \times 10{^2}{^8}\) free electrons per cubic meter, find the average drift speed under the conditions of part (b).

On your first day at work as an electrical technician, you are asked to determine the resistance per meter of a long piece of wire. The company you work for is poorly equipped. You find a battery, a voltmeter, and an ammeter, but no meter for directly measuring resistance (an ohmmeter). You put the leads from the voltmeter across the terminals of the battery, and the meter reads 12.6 V. You cut off a 20.0-m length of wire and connect it to the battery, with an ammeter in series with it to measure the current in the wire. The ammeter reads 7.00 A. You then cut off a 40.0-m length of wire and connect it to the battery, again with the ammeter in series to measure the current. The ammeter reads 4.20 A. Even though the equipment you have available to you is limited, your boss assures you of its high quality: The ammeter has very small resistance, and the voltmeter has very large resistance. What is the resistance of 1 meter of wire?

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