Problem 8 - College Physics

Question

According to the Bohr model of the hydrogen atom (see Chapter 27 ), the electron can exist only in circular orbits of certain radii about a proton. (a) Will a larger orbit have (1) a higher, (2) an equal, or (3) a lower electric potential than a smaller orbit? Why? (b) Determine the potential difference between two orbits of radii \(0.21 \mathrm{nm}\) and \(0.48 \mathrm{nm}\).

Step-by-Step Solution

Answer

(a) Lower; (b) Potential difference is 3.84 V.

Step 1: Understanding the Electric Potential Concept

The electric potential \( V \) at a distance \( r \) from a point charge \( Q \) is given by the formula \( V = \frac{kQ}{r} \), where \( k \) is Coulomb's constant. For a hydrogen atom, the proton acts as a point charge. Since the electric potential is inversely proportional to the radius \( r \), a larger orbit (greater \( r \)) will have a lower electric potential.

Step 2: Calculating the Potential Difference

To find the potential difference \( \Delta V \) between two orbits, we use the formula \( \Delta V = V_i - V_f = \frac{kQ}{r_i} - \frac{kQ}{r_f} \). Let's calculate this for the given radii \( r_i = 0.21 \text{ nm} = 0.21 \times 10^{-9} \text{ m} \) and \( r_f = 0.48 \text{ nm} = 0.48 \times 10^{-9} \text{ m} \). Assuming the charge of the proton \( Q = 1.6 \times 10^{-19} \text{ C} \) and Coulomb's constant \( k = 8.99 \times 10^9 \text{ Nm}^2/ ext{C}^2 \), we calculate the potential difference.

Step 3: Substituting Values and Solving

Substitute the values into the formula:\[\Delta V = \frac{8.99 \times 10^9 \times 1.6 \times 10^{-19}}{0.21 \times 10^{-9}} - \frac{8.99 \times 10^9 \times 1.6 \times 10^{-19}}{0.48 \times 10^{-9}}\]Simplifying the expression, calculate each term separately:\[V_i = \frac{8.99 \times 10^9 \times 1.6 \times 10^{-19}}{0.21 \times 10^{-9}} \approx 6.84 \text{ V}\]\[V_f = \frac{8.99 \times 10^9 \times 1.6 \times 10^{-19}}{0.48 \times 10^{-9}} \approx 3.00 \text{ V}\]Thus, the potential difference \( \Delta V \) is approximately:\[\Delta V = 6.84 \text{ V} - 3.00 \text{ V} = 3.84 \text{ V}\]

Step 4: Conclusion

For part (a), a larger orbit has a lower electric potential than a smaller orbit due to the inverse relationship between potential and radius. For part (b), the potential difference between the two orbits with radii \(0.21 \text{ nm}\) and \(0.48 \text{ nm}\) is \(3.84 \text{ V}\).

Key Concepts

Hydrogen Atom

The hydrogen atom is the simplest atom and serves as a fundamental model for understanding atomic structure. It consists of one proton and one electron. In the Bohr model, the electron orbits the proton in specific circular paths at distinct energy levels. These paths are known as shells or orbits. Each orbit has a fixed radius and an associated energy level.
The Bohr model of the hydrogen atom explains that electrons can only occupy orbits where the angular momentum is quantized. This means that the angular momentum takes on specific, discrete values.
Electrons transition between these orbits by absorbing or emitting energy in the form of light. When moving to a higher orbit, an electron absorbs energy; when moving to a lower orbit, it emits energy. This model helps to understand not only the energy levels within an atom but also the spectral lines associated with the hydrogen atom.

Electric Potential

Electric potential, often denoted by \( V \), is the potential energy per unit charge associated with a position in an electric field. It's a scalar quantity, meaning it only has a magnitude and no direction. In the context of atoms, it refers to the potential energy experienced by an electron due to the nucleus.
Mathematically, the electric potential due to a point charge \( Q \) at a distance \( r \) is given by the formula:

• \[ V = \frac{kQ}{r} \]

where \( k \) is Coulomb's constant \((8.99 \times 10^9 \, \mathrm{Nm^2/C^2})\).
In a hydrogen atom, since the proton behaves as a point charge, the potential encountered by an electron at a distance \( r \) from the proton can be derived using the same formula. The electric potential is inversely related to the distance \( r \), meaning it decreases as the radius increases. This concept is crucial for understanding the behavior of electrons in the Bohr model.

Potential Difference

The potential difference, also known as voltage, is a measure of the change in electric potential between two points. It tells us how much work is needed to move a charge from one point to another within an electric field. In a physics context like the Bohr model of the hydrogen atom, potential difference plays a key role in determining energy transitions between orbits.
To calculate the potential difference \( \Delta V \) between two orbits, you can use the formula:

• \[ \Delta V = \frac{kQ}{r_i} - \frac{kQ}{r_f} \]

where \( r_i \) and \( r_f \) are the initial and final orbit radii, respectively.
This expression is simply the difference between the electric potential at two different radii. Understanding potential difference is essential for analyzing how much work would be required to move electrons between orbits in an atom.

Coulomb's Law

Coulomb's law is a fundamental principle that describes the force between two charged particles. It's essential for understanding interactions in the Bohr model of the hydrogen atom. According to Coulomb's law, the force \( F \) between two charges is directly proportional to the product of their magnitudes and inversely proportional to the square of their distance apart. Mathematically, it is expressed as:

• \[ F = \frac{k |Q_1 Q_2|}{r^2} \]

where \( k \) is Coulomb’s constant, \( Q_1 \) and \( Q_2 \) are the magnitudes of the charges, and \( r \) is the distance between the charges.
In the hydrogen atom, Coulomb’s law explains the attractive force between the negatively charged electron and the positively charged proton. This force keeps the electron in its orbit around the nucleus. Without this force, the electron would not be confined to specific orbits, and the Bohr model would not apply. Understanding Coulomb's law provides insight into the stability of atomic structures and the basis for calculating electric potential in atomic systems.