/*! 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 13 An electron and a proton are ini... [FREE SOLUTION] | 91影视

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Chapter 19: Problem 13

An electron and a proton are initially very far apart (effectively an infinite distance apart). They are then brought together to form a hydrogen atom, in which the electron orbits the proton at an average distance of \(5.29 \times 10^{-11} \mathrm{~m}\). What is EPE \(_{\text {final }}-\) EPE \(_{\text {initial }}\), which is the change in the electric potential energy?

Short Answer

Expert verified
The change in electric potential energy is approximately \(-2.18 \times 10^{-18} \mathrm{~J} \).

Step by step solution

01

Understand the Problem

We are asked to find the change in electric potential energy when an electron and a proton come together to form a hydrogen atom. Initially, they are very far apart, and then they orbit with an average distance.
02

Identify the Formula

The electric potential energy (EPE) for two point charges is given by: \[EPE = \frac{k \cdot q_1 \cdot q_2}{r}\] where \(k = 8.99 \times 10^9 \mathrm{~N \, m^2/C^2}\) is the Coulomb's constant, \(q_1\) and \(q_2\) are the charges of the electron and proton respectively, and \(r\) is the distance between the charges.
03

Input Known Values

The charge of an electron \( (q_1) \) is \(-1.6 \, \times 10^{-19} \mathrm{~C}\) and the charge of a proton \( (q_2) \) is \(1.6 \, \times 10^{-19} \mathrm{~C}\). The average distance between them \(r\) in a hydrogen atom is \(5.29 \times 10^{-11} \, \mathrm{m}\).
04

Calculate EPE_initial

Initially, the electron and proton are very far apart, so we consider \( \mathrm{EPE}_{\text{initial}} \) to be zero, as potential energy at an infinite distance is zero.
05

Calculate EPE_final

Input the values into the formula for EPE: \[\mathrm{EPE}_{\text{final}} = \frac{(8.99 \times 10^9) \cdot (-1.6 \times 10^{-19}) \cdot (1.6 \times 10^{-19})}{5.29 \times 10^{-11}}\] Calculate to find \( \mathrm{EPE}_{\text{final}}\).
06

Compute the Difference in EPE

The change in electric potential energy is: \[\mathrm{EPE}_{\text{final}} - \mathrm{EPE}_{\text{initial}} = \mathrm{EPE}_{\text{final}} - 0 = \mathrm{EPE}_{\text{final}}\] Perform the calculation to determine the final answer.
07

Final Calculation

When you plug in the numbers from Step 5, the final calculation for \( \mathrm{EPE}_{\text{final}} \) gives approximately \[-2.18 \times 10^{-18} \mathrm{~J} \]. Thus, this value represents the change in electric potential energy.

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Key Concepts

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

Coulomb's constant
Coulomb's constant, often represented as \( k \), is a crucial value in electrostatics that defines the strength of the electric force between two point charges. The constant is valued at approximately \( 8.99 \times 10^9 \, \mathrm{N \cdot m^2/C^2} \). This constant is a part of Coulomb's Law equation, which allows us to calculate the electric force or electric potential energy between two charges.
  • It helps quantify how much force two charges exert on each other.
  • The constant arises from the permittivity of free space and is derived for systems in a vacuum.
  • Its value is crucial for calculations involving electric interactions like those in a hydrogen atom.
Coulomb's constant is used to calculate electric potential energy, which is a measure of the work needed to bring charges from an infinite distance apart to a specific separation distance.
Hydrogen atom
The hydrogen atom is the simplest atom in the universe, consisting of just one proton and one electron. In this exercise, we analyze a hydrogen atom in terms of electric potential energy.
  • The electron orbits the proton at an average distance known as the Bohr radius, approximately \( 5.29 \times 10^{-11} \, \mathrm{m} \).
  • Despite its simplicity, the hydrogen atom is vital for understanding atomic structure and quantum mechanics.
In a hydrogen atom, the electron-proton interaction can be quantitatively described using the concepts of electric force and electric potential energy, highlighting the role of Coulomb's law in atomic physics.
Electron-proton interaction
The electron-proton interaction in a hydrogen atom is a fundamental electrostatic force. It is the attractive force that holds the electron in orbit around the proton.
  • This interaction is driven by the opposite charges of the electron and proton.
  • It keeps the electron in the lowest potential energy state around the nucleus.
  • This interaction forms the basis for many atomic behaviors and is essential in calculating electric potential energy.
Understanding this interaction is crucial for analyzing how atoms function and form the building blocks for chemistry and solids.
Charge of electron and proton
The charges of an electron and a proton are fundamental properties of these particles. The electron carries a negative charge, while the proton carries a positive charge, and these charges are equal in magnitude but opposite.
  • The electronic charge for the electron is \( -1.6 \times 10^{-19} \, \mathrm{C} \).
  • The charge of a proton is \( 1.6 \times 10^{-19} \, \mathrm{C} \).
  • The interaction of these charges results in the electrostatic attraction that forms the hydrogen atom.
These charges are the foundation for calculating electric forces and electric potential energy, influencing interactions within and between atoms.

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Most popular questions from this chapter

Identical point charges of \(+1.7 \mu \mathrm{C}\) are fixed to diagonally opposite corners of a square. A third charge is then fixed at the center of the square, such that it causes the potentials at the empty corners to change signs without changing magnitudes. Find the sign and magnitude of the third charge.

Concept Questions Two capacitors have the same plate separation. However, one has square plates, while the other has circular plates. The square plates are a length \(L\) on each side, and the diameter of the circular plates is \(L\). (a) If the same dielectric material were between the plates in each capacitor, which one would have the greater capacitance? (b) By putting different dielectric materials between the capacitor plates, we can make the two capacitors have the same capacitance. Which capacitor should contain the dielectric material with the greater dielectric constant? Give your reasoning in each case. The capacitors have the same capacitance because they contain different dielectric materials. The dielectric constant of the material between the square plates has a value of \(\kappa_{\text {square }}=3.00 .\) What is the dielectric constant \(\kappa_{\text {circle }}\) of the material between the circular plates? Be sure that your answer is consistent with your answers to the Concept Questions.

Refer to Interactive Solution 19.45 at and Multiple-Concept Example 10 for help in solving this problem. An empty capacitor has a capacitance of \(3.2 \mu \mathrm{F}\) and is connected to a 12-V battery. A dielectric material \((\kappa=4.5)\) is inserted between the plates of this capacitor. What is the magnitude of the surface charge on the dielectric that is adjacent to either plate of the capacitor? (Hint: The surface charge is equal to the difference in the charge on the plates with and without the dielectric.)

The membrane that surrounds a certain type of living cell has a surface area 8 of \(5.0 \times 10^{-9} \mathrm{~m}^{2}\) and a thickness of \(1.0 \times 10^{-8} \mathrm{~m}\). Assume that the membrane behaves like a parallel plate capacitor and has a dielectric constant of \(5.0 .\) (a) The potential on the outer surface of the membrane is \(+60.0 \mathrm{mV}\) greater than that on the inside surface. How much charge resides on the outer surface? (b) If the charge in part (a) is due to \(\mathrm{K}^{+}\) ions (charge \(+e\) ), how many such ions are present on the outer surface?

An axon is the relatively long tail-like part of a neuron, or nerve cell. The outer surface of the axon membrane (dielectric constant \(=5,\) thickness \(=1 \times 10^{-8} \mathrm{~m}\) ) is charged positively, and the inner portion is charged negatively. Thus, the membrane is a kind of capacitor. Assuming that an axon can be treated like a parallel plate capacitor with a plate area of \(5 \times 10^{-6} \mathrm{~m}^{2},\) what is its capacitance?
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