Problem 15 - College Physics

Question

\(\bullet$$\bullet\) Two small aluminum spheres, each having mass 0.0250 \(\mathrm{kg}\) , are separated by 80.0 \(\mathrm{cm}\) . (a) How many electrons does each sphere contain? (The atomic mass of aluminum is 26.982 \(\mathrm{g} / \mathrm{mol}\) , and its atomic number is \(13 .\) (b) How many electrons would have to be removed from one sphere and added to the other to cause an attractive force between the spheres of magnitude \(1.00 \times 10^{4} \mathrm{N}\) (roughly 1 ton)? Assume that the spheres may be treated as point charges. (c) What fraction of all the elec- trons in each sphere does this represent?

Step-by-Step Solution

Answer

(a) Each sphere contains approximately \(7.2 \times 10^{24}\) electrons. (b) \(1.5 \times 10^{13}\) electrons need to be moved. (c) The fraction is extremely small, on the order of \(10^{-11}\).

Step 1: Calculate Moles of Aluminum in a Sphere

The mass of each sphere is 0.025 kg, which is equivalent to 25 g. The atomic mass of aluminum is 26.982 g/mol. To find the moles of aluminum in a sphere, we use the formula \( \text{moles} = \frac{\text{mass}}{\text{molar mass}} \): \[ \text{moles of } \text{Al} = \frac{25\, \text{g}}{26.982\, \text{g/mol}} \approx 0.925 \text{ moles} \].

Step 2: Calculate Number of Electrons in Each Sphere

Since aluminum has an atomic number of 13, each aluminum atom has 13 electrons. The number of electrons in one mole of any substance is Avogadro's number, \(6.022 \times 10^{23}\). So, the number of electrons in each sphere is: \[ \text{Number of electrons} = 0.925 \text{ moles} \times 13 \text{ electrons/mole} \times 6.022 \times 10^{23} \text{ electrons/mole} \].

Step 3: Calculate Charge Needed for Given Force

Using Coulomb's Law, the force \( F = k \frac{|q_1 q_2|}{r^2} \), where \( r = 0.8 \text{ m} \) is the separation, \( k = 8.99 \times 10^9 \text{ N·m}^2/\text{C}^2 \), and \( F = 1 \times 10^4 \text{ N} \). Assume the charge on each sphere is equal and opposite, \( q_1 = -q_2 = q \). The formula becomes: \[ 1 \times 10^{4} = k \frac{q^2}{(0.8)^2} \],Solve for \( q \).

Step 4: Number of Electrons to Move to Produce the Charge

Once you have computed the charge \( q \) from Step 3, use the relation \( q = n \times e \), where \( e = 1.602 \times 10^{-19} \text{ C} \) is the charge of one electron. Solve for \( n \), the number of electrons: \[ n = \frac{q}{e} \].

Step 5: Calculate the Fraction of Electrons Moved

Use the result from Step 4 to find the fraction of electrons moved. This is given by: \[ \text{fraction} = \frac{n}{\text{Total number of electrons in a sphere}} \].Insert the total number of electrons from Step 2, and the number moved from Step 4.

Key Concepts

Coulomb's Law

Coulomb's Law is a fundamental principle in electrostatics that describes the force between two charged objects. The law can be expressed by the equation: \[ F = k \frac{|q_1 q_2|}{r^2} \]where:

• \( F \) is the magnitude of the force between the charges.
• \( q_1 \) and \( q_2 \) are the magnitudes of the charges on the two objects.
• \( r \) is the distance between the centers of the two charges.
• \( k \) is the Coulomb's constant, approximately \( 8.99 \times 10^9 \text{ N·m}^2/\text{C}^2 \).

This law tells us that the force between two charges is proportional to the product of their charges and inversely proportional to the square of the distance between them. If charges are of opposite signs, the force will be attractive, pulling the objects together. If they are the same, the force will be repulsive, pushing them apart. By applying Coulomb's Law, we can determine how strong the electrostatic force will be for any given arrangement of charges.

Atomic Structure

The atomic structure refers to the composition of an atom, which is the basic unit of matter. An atom consists of three main types of particles:

• Protons: Positively charged particles found in the nucleus.
• Neutrons: Neutral particles that also reside in the nucleus.
• Electrons: Negatively charged particles that move around the nucleus.

The number of protons in the nucleus defines the element and is known as the atomic number. For aluminum, this number is 13. This means each aluminum atom has 13 protons. Correspondingly, in a neutral aluminum atom, there are also 13 electrons, which help balance the positive charge of the protons.
The arrangement of electrons around the nucleus follows specific patterns or shells. These electrons play a critical role in determining how atoms interact and bond with each other, influencing the material's chemical and physical properties. Understanding atomic structure is crucial for explaining how substances behave and react in various situations.

Electron Transfer

Electron transfer is a process where electrons move from one atom or molecule to another. This movement of electrons is fundamental in chemical reactions and is essential in the formation of ionic bonds. In the context of electrostatics, transferring electrons between objects can induce a net charge and generate an electrostatic force, as seen in the exercise at hand. When we discuss electron transfer between the two aluminum spheres in the original problem, we are focusing on modifying the net charge of these spheres to create a force. The transfer of electrons from one sphere to another changes their charge balance as:

• The sphere losing electrons becomes positively charged.
• The sphere gaining electrons becomes negatively charged.

This difference in charge leads to an electrostatic force, which can be calculated using Coulomb's Law. By controlling the number of electrons transferred, we can adjust the amount of force exerted between the charged spheres.