/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 74 A \(100 \mathrm{~W}\) lamp has a... [FREE SOLUTION] | 91影视

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Chapter 21: Problem 74

A \(100 \mathrm{~W}\) lamp has a steady current of \(0.83 \mathrm{~A}\) in its filament. How long is required for 1 mol of electrons to pass through the lamp?

Short Answer

Expert verified
It takes approximately 116235 seconds for 1 mole of electrons to pass through the lamp.

Step by step solution

01

Understand the Problem

We need to find out how long it takes for 1 mole of electrons to pass through a lamp with a steady current of 0.83 A.
02

Use the Formula for Current

Current is defined as the flow of charge per time. The formula for current is given by the equation: \[ I = \frac{Q}{t} \]where \(I\) is the current, \(Q\) is the electric charge, and \(t\) is the time. We need to solve for \(t\), so we rearrange the equation as \[ t = \frac{Q}{I} \].
03

Calculate the Charge for 1 Mole of Electrons

1 mole of electrons contains Avogadro's number of electrons, which is approximately \(6.022 \times 10^{23}\) electrons. The charge of one electron is approximately \(1.602 \times 10^{-19}\) coulombs. Therefore, the total charge \(Q\) of 1 mole of electrons is: \[ Q = 6.022 \times 10^{23} \times 1.602 \times 10^{-19} \]\[ Q \approx 96485 \text{ C} \]This value is also known as the Faraday constant.
04

Calculate the Time

Now that we have the charge, we can substitute \(Q\) into the rearranged current formula:\[ t = \frac{96485 \text{ C}}{0.83 \text{ A}} \]Calculating this gives:\[ t \approx 116235 \text{ seconds} \].

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Key Concepts

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

Electric Charge
Electric charge is a fundamental property of matter. It's like a small packet of energy that particles carry. When we talk about electric charge, we often refer to two types: positive and negative. These charges create electric fields and currents, impacting how particles interact with each other.
  • Electrons have a negative charge of approximately \( -1.602 \times 10^{-19} \) coulombs.
  • Protons, on the other hand, have a positive charge of the same magnitude.
When electrons flow through a conductor, like the filament of a lamp, they create an electric current. The measure of this flow is given in amperes (A), where 1 ampere equals 1 coulomb of charge flowing past a point per second.
Avogadro's Number
Avogadro's number is a huge number that helps us count particles, like atoms and molecules, in chemistry. This number is approximately \( 6.022 \times 10^{23} \), representing how many atoms, ions, or molecules are in one mole of a substance.
  • Used widely in both chemistry and physics.
  • Helps connect the macroscopic world with the microscopic world.
For example, when we're dealing with electrons, one mole of electrons contains Avogadro's number of them. This concept is crucial when calculating the total charge for a specific number of electrons.
Faraday Constant
The Faraday constant is a physical constant that represents the charge of one mole of electrons. It's approximately equal to \( 96485 \) coulombs per mole.
  • This value stems from multiplying Avogadro's number by the charge of a single electron.
  • It's widely used in electrochemistry when calculating electric charge.
Understanding the Faraday constant provides insight into the bridge between electric charge at the molecular level and macroscopic measurements in experiments. This constant shows the immense number of electrons involved to produce a measurable electric charge.
Time Calculation
Time calculation in the context of electric circuits involves understanding how long it takes for a certain amount of charge to pass through a conductor. To calculate this, we often use the formula: \[ I = \frac{Q}{t} \]Where:
  • \( I \) is the current in amperes.
  • \( Q \) is the charge in coulombs.
  • \( t \) is the time in seconds.
By rearranging the formula to \( t = \frac{Q}{I} \), we can solve for the time. For example, to find how long it takes for 1 mole of electrons to pass through a lamp with a 0.83 A current, we calculate using the Faraday constant as the electric charge. The final time, \(116235\) seconds, shows that time calculations require a thorough understanding of both charge and current.

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Most popular questions from this chapter

Figure \(21-37\) shows four identical conducting spheres that are actually well separated from one another. Sphere \(W\) (with an initial charge of zero) is touched to sphere \(A\) and then they are separated. Next, sphere \(W\) is touched to sphere \(B\) (with an initial charge of \(-32 e\) ) and then they are separated. Finally, sphere \(W\) is touched to sphere \(C\) (with an initial charge of \(+48 e\) ), and then they are separated. The final charge on sphere \(W\) is \(+18 e\). What was the initial charge on sphere \(A\) ?

What must be the distance between point charge \(q_{1}=\) \(26.0 \mu \mathrm{C}\) and point charge \(q_{2}=-47.0 \mu \mathrm{C}\) for the electrostatic force between them to have a magnitude of \(5.70 \mathrm{~N} ?\)

Earth's atmosphere is constantly bombarded by cosmic ray protons that originate somewhere in space. If the protons all passed through the atmosphere, each square meter of Earth's surface would intercept protons at the average rate of 1500 protons per second. What would be the electric current intercepted by the total surface area of the planet?

Two particles are fixed on an \(x\) axis. Particle 1 of charge \(40 \mu \mathrm{C}\) is located at \(x=-2.0 \mathrm{~cm} ;\) particle 2 of charge \(Q\) is located at \(x=3.0 \mathrm{~cm} .\) Particle 3 of charge magnitude 20 \(\mu \mathrm{C}\) is released from rest on the \(y\) axis at \(y=2.0 \mathrm{~cm}\). What is the value of \(O\) if the initial acceleration of particle 3 is in the positive direction of (a) the \(x\) axis and (b) the \(y\) axis?

How many electrons would have to be removed from a coin to leave it with a charge of \(+1.0 \times 10^{-7} \mathrm{C}\) ?
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