Problem 12 - College Physics

Question

\(\bullet\) (a) What is the total negative charge, in coulombs, of all the electrons in a small 1.00 g sphere of carbon? One mole of C is \(12.0 \mathrm{g},\) and each atom contains 6 protons and 6 electrons. (b) Suppose you could take out all the electrons and hold them in one hand, while in the other hand you hold what is left of the original sphere. If you hold your hands 1.50 \(\mathrm{m}\) apart at arms length, what force will each of them feel? Will it be attractive or repulsive?

Step-by-Step Solution

Answer

(a) The total negative charge is approximately \(-4.82 \times 10^4\ \text{C}\). (b) The force is about \(9.25 \times 10^{19}\ \text{N}\), and it is attractive.

Step 1: Calculate the number of moles of carbon

Since we have 1.00 gram of carbon and the molar mass of carbon is 12.0 g/mol, we calculate the number of moles by dividing the mass by the molar mass:\[\text{Number of moles} = \frac{1.00\ \text{g}}{12.0\ \text{g/mol}} = 0.0833\ \text{mol}\]

Step 2: Determine the number of atoms

One mole of carbon contains Avogadro's number of atoms, which is approximately \(6.022 \times 10^{23}\) atoms/mol. So, the number of carbon atoms in 0.0833 mol is:\[\text{Number of atoms} = 0.0833\ \text{mol} \times 6.022 \times 10^{23}\ \text{atoms/mol} = 5.016\times 10^{22}\ \text{atoms}\]

Step 3: Calculate the total number of electrons

Each carbon atom has 6 electrons. Therefore, the total number of electrons is:\[\text{Total electrons} = 5.016 \times 10^{22}\ \text{atoms} \times 6\ \text{electrons/atom} = 3.010\times 10^{23}\ \text{electrons}\]

Step 4: Calculate the total negative charge

The charge of a single electron is approximately \(-1.6 \times 10^{-19}\) coulombs. Thus, the total charge is:\[\text{Total charge} = 3.010 \times 10^{23}\ \text{electrons} \times (-1.6 \times 10^{-19}\ \text{C/electron}) = -4.82 \times 10^{4}\ \text{C}\]

Step 5: Calculate the electrostatic force

The force between two charges can be calculated using Coulomb's Law: \[ F = \frac{k \cdot |q_1 \cdot q_2|}{r^2} \]where \(k\) is Coulomb's constant \(8.988 \times 10^9\ \text{N m}^2/\text{C}^2\), \(q_1\) and \(q_2\) are the charges, and \(r\) is the distance between the charges. Here, \(q_1 = 4.82 \times 10^4\ \text{C}\), \(q_2\) is the opposite charge of the same magnitude, and \(r = 1.50\ \text{m}\):\[ F = \frac{8.988 \times 10^9\ \text{N m}^2/\text{C}^2 \times (4.82 \times 10^4\ \text{C})^2}{(1.50\ \text{m})^2} \]Calculating gives:\[ F \approx 9.25 \times 10^{19}\ \text{N}\]

Step 6: Determine the nature of the force

Since one hand holds the electrons (negative charge) and the other holds the remainder of the sphere (positive charge), the force will be attractive.

Key Concepts

Coulomb's Law

Coulomb's Law is a fundamental principle in electrostatics that describes the force between two stationary electric charges. It states that the force (\( F \)) between two point charges (\( q_1 \) and \( q_2 \)) is directly proportional to the product of the magnitudes of the charges and inversely proportional to the square of the distance (\( r \)) between them. The law can be expressed mathematically as:
\[ F = \frac{k \cdot |q_1 \cdot q_2|}{r^2} \]
where \( k \) is Coulomb's constant, approximately \( 8.988 \times 10^9\ \text{N m}^2/\text{C}^2 \).

**Key Points**

• The force is attractive if the charges are of opposite signs, and repulsive if they are of the same sign.
• Coulomb's Law shows that the force is quite powerful over very small distances.
• The law applies to point charges, where the size of the charges is negligible compared to the separation distance.

In the given problem, after calculating the charge equivalent to holding electrons in one hand and protons in the other, we use Coulomb's Law to find out that the force at a distance of 1.5 meters is overwhelmingly large, calculated to be about \( 9.25 \times 10^{19}\ \text{N} \), and it is attractive because of the opposite nature of the charges.

Electric Charge

Electric charge is a fundamental property of particles that causes them to experience a force when placed in an electromagnetic field. There are two types of electric charges: positive and negative. Particles with the same type of charge repel each other, while particles with opposite charges attract each other.

**Units and Measurement**

• The unit of electric charge in the International System of Units (SI) is the coulomb (C).
• One coulomb is equivalent to the total charge transferred by a constant current of one ampere in one second.
• The elementary charge, denoted by \( e \), is the smallest unit of charge, carried by a single proton or electron.

Electrons have a charge of approximately \(-1.6 \times 10^{-19}\ \text{C}\), while protons have a charge of \(+1.6 \times 10^{-19}\ \text{C}\). In our problem, the negative charge from electrons in a 1 g carbon sphere is calculated to be \(-4.82 \times 10^4\ \text{C}\), showcasing how numerous these tiny charges are when accumulated.

Avogadro's Number

Avogadro's Number is a defining constant in chemistry and physics that denotes the number of constituent particles (usually atoms or molecules) in one mole of a given substance. Avogadro's Number is approximately \( 6.022 \times 10^{23}\), symbolized by \( N_A \). This number allows scientists to relate macroscopic quantities of a substance to the number of atoms or molecules, thereby bridging the gap between the atomic scale and everyday quantities.

**Applications in Calculations**

• When dealing with problems involving moles, Avogadro's Number helps in converting between the mass of a substance and the number of atoms present.
• It is instrumental in various chemistry calculations such as determining number of atoms in a sample.

In the context of the carbon sphere problem, Avogadro's Number was used to find the total number of atoms and, consequently, electrons in the 1 g sample of carbon. This crucial step allowed us to determine the total negative charge contained in those electrons.

Carbon Atom

The carbon atom is an essential element in chemistry and life sciences, known for its ability to form stable bonds with many elements, including itself. This versatility is why carbon is a central element in organic chemistry.

**Structure of a Carbon Atom**

• A carbon atom contains 6 protons and 6 neutrons within its nucleus, and is typically surrounded by 6 electrons.
• The electrons are arranged in shells around the nucleus, with 2 electrons in the inner shell and 4 in the outer shell, leading to carbon's valency of four.

Carbon's place on the periodic table makes it uniquely suited to form complex and diverse molecules.

In our exercise, understanding that each carbon atom contains 6 electrons helps to calculate the total negative charge in a 1-gram carbon sample. This clear, foundational knowledge of the carbon atom's structure is pivotal in solving many chemical and physical problems.