Question

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

The current is approximately 0.752 mA, directed from A to B.

Step-by-step solution

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 \(-5.11 \times 10^{18} \times 1.6 \times 10^{-19} \text{ C}\).

Calculate Charge of Ca2+ ions

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

Calculate Total Charge Exchange

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

Convert Time Units

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

Calculate Current

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

Determine Current Direction

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

Key concepts

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.