Question

How many electrons flow when a current of \(5 \mathrm{~A}\) is passed through a solution for \(200 \mathrm{~s}\) ? (a) \(6.022 \times 10^{23}\) (b) \(6.24 \times 10^{21}\) (c) \(6.024 \times 10^{21}\) (d) \(6.022 \times 10^{20}\)

Short answer

Approximately \(6.24 \times 10^{21}\) electrons flow through the solution.

Step-by-step solution

Understand the relationship between current, charge, and time

The current (I) is the flow of electric charge (Q) per unit time (t). This relationship is given by the formula: \( I = \frac{Q}{t} \). To find the total charge, we rearrange the formula to \( Q = I \cdot t \).

Calculate the total charge that flows

Using the given current of \( 5 \) A and time of \( 200 \) s, we calculate the total charge as follows: \( Q = 5 \, \mathrm{A} \cdot 200 \, \mathrm{s} = 1000 \, \mathrm{C} \) (where C is Coulombs).

Convert charge to number of electrons

The charge of one electron is approximately \( 1.602 \times 10^{-19} \) C. To find the number of electrons that corresponds to \( 1000 \, \mathrm{C} \), we divide the total charge by the charge of one electron: \( \text{Number of electrons} = \frac{1000 \, \mathrm{C}}{1.602 \times 10^{-19} \, \mathrm{C/e^{-}}} \approx 6.24 \times 10^{21} \) electrons.

Key concepts

Current and Charge Relationship

Understanding the relationship between electric current and charge is fundamental in the study of electricity. Electric current can be thought of as the flow of electric charge through a conductor. This current is measured in amperes (A), which is essentially the rate at which the charge flows. More formally, one ampere is defined as one coulomb of charge passing through a given point in a circuit per second.

The link between current (I), charge (Q), and time (t) is given by the equation: \[ I = \frac{Q}{t} \.\] Here, if you want to solve for the charge transferred over a period of time, you can rearrange the equation as \[ Q = I \times t \]. By multiplying the current by the time interval during which the current flows, you get the total charge that has moved through the circuit.

This equation is vital when you are dealing with any problem that involves calculating the movement of charge in a circuit over time. In scenarios where you are given the current and the duration for which it flows, you can easily calculate the total charge transferred. This relationship serves as the foundation for many practical applications, including the calculation of electrical energy consumption and designing electrical circuits.

Calculating Electron Flow

Once you have calculated the total charge that flows in a circuit using the relationship between current and charge, the next concept involves understanding and calculating the flow of electrons. Electrons are the tiny particles that carry charge through circuits, and their flow constitutes the electric current. To determine the number of electrons that correspond to a certain amount of charge, we use the fundamental charge of an electron, which is approximately \(1.602 \times 10^{-19}\) coulombs (C).

To calculate the number of electrons flowing in a circuit, the following formula is used: \[ \text{Number of electrons} = \frac{Q}{e} \], where \( e \) is the charge of one electron. By dividing the total charge (\( Q \) in coulombs) by the charge of a single electron, you'll get the number of electrons that have flown through the circuit. This calculation is crucial for understanding the microscopic behavior of electric currents, which is predominantly the flow of electrons in a conductor.

For example, if a device uses a current of \(5\) A for \(200\) s, meaning a total charge of \(1000\) C flows through it, the number of electrons flowing can be calculated by dividing \(1000\) C with \(1.602 \times 10^{-19}\) C to get approximately \(6.24 \times 10^{21}\) electrons.

Charge and Time Formula

The charge and time formula, \( Q = I \times t \), is a powerful tool in the field of electrical engineering and physics. It simply states that the charge transferred through a certain point in a circuit over a given time interval can be calculated by multiplying the current by the time. This formula is especially useful when solving problems related to battery life, capacitor charging and discharging, and even in the realm of electrochemistry.

In practical terms, suppose you need to calculate the lifespan of a battery. By knowing the battery's current output and the total charge it can provide, you can apply the charge and time formula to determine how long the battery will last. Similarly, this formula facilitates the calculation of how long it will take to charge a capacitor to a certain voltage when the current is known.

To enhance the students understanding, it's often helpful to think of the formula in terms of a water analogy: current is the rate of water flow, charge is the amount of water that flows, and time is the duration the water flows for. Just as a certain volume of water flows through a pipe over a certain time at a given rate, the electric charge flows through a circuit over a certain time at a given current. This visualization can make the concept more relatable and easier to grasp.