Question

For each of the questions, four choices have been provided. Select the correct alternative. What is the maximum number of electrons, present in an orbit whose angular momentum is \(\frac{2 \mathrm{~h}}{\pi}\) ? (a) 18 (b) 2 (c) 32 (d) 8

Short answer

Answer: The maximum number of electrons in the orbit is 8.

Step-by-step solution

Recall Bohr's Quantization Rule

According to Bohr's quantization rule for angular momentum, the angular momentum of an electron in an orbit \((L)\) is given by: \(L = mvr = \frac {nh}{2\pi}\) Where \(m\) is the mass of the electron, \(v\) is its velocity, \(r\) is the radius of the orbit, \(n\) is the principal quantum number, and \(h\) is the Planck's constant.

Find the principal quantum number (n)

In this exercise, we are given the angular momentum of the orbit as \(\frac{2h}{\pi}\). We will use Bohr's quantization rule to find the principal quantum number by equating the given angular momentum with the formula: \(\frac{2h}{\pi} = \frac {nh}{2\pi}\) Dividing both sides by \(h\) and multiplying by \(2\pi\), we obtain: \(2 = n\) So, the principal quantum number (\(n\)) for this orbit is 2.

Determine the maximum number of electrons

The maximum number of electrons in an orbit can be determined using the formula: \(max~electrons = 2n^2\) Substituting the value of \(n\) we found in step 2, we get: \(max~electrons = 2(2^2) = 2(4) = 8\) The maximum number of electrons in the orbit is 8, which corresponds with choice (d) in the exercise.

Key concepts

Understanding Angular Momentum

Angular momentum is a fundamental concept in physics, crucial for understanding electron behavior in atoms. In Bohr's model of the atom, angular momentum quantization means that electrons can only occupy specific orbits. These discrete orbits arise because the angular momentum is quantized. This quantization is expressed as:\[ L = \frac{nh}{2\pi} \]Where:

• \(L\) is the angular momentum.
• \(n\) is the principal quantum number.
• \(h\) is Planck's constant.

This formula shows that angular momentum depends on the principal quantum number, \(n\), which determines the electron's orbit. For instance, an orbit with \(L = \frac{2h}{\pi}\) implies a principal quantum number \(n\) of 2. Consequently, this particular orbit dictates specific properties, such as allowed electron numbers. Ideal for young learners, these quantized levels help visualize electron placements like planets in the solar system—specific, predictable, and stable.

Principal Quantum Number Simplified

The principal quantum number, \(n\), is a key quantum mechanical description of an electron in an atom. It not only defines the size and energy level of the orbit but also indirectly determines other properties like angular momentum. The principal quantum number is always a positive integer (1, 2, 3, ...).In our current example, where the angular momentum is given as \(\frac{2h}{\pi}\), solving Bohr's quantization rule results in \(n = 2\). This \(n\) value situates the electron in the second energy level or shell.Key takeaways about the principal quantum number:

• Directly influences energy levels, with higher \(n\) increasing energy.
• Helps define the number of subshells within each energy level.
• Affects the maximum number of electrons in the orbit through the formula \(2n^2\).

Understanding \(n\) thus grounds us in determining energy dynamics and electron configurations within atoms.

Calculating Maximum Number of Electrons

The maximum number of electrons that an orbit can hold is determined by the principal quantum number, \(n\), using the formula:\[ max\_electrons = 2n^2 \]Here, \(n = 2\), so substituting this value gives:\[ max\_electrons = 2 \times (2^2) = 2 \times 4 = 8 \]This shows that the highest number of electrons that can be present in this specific orbit is 8. More generally, the equation \(2n^2\) provides a quick way to ascertain how many electrons any given shell can contain:

• \(n=1\): Can hold 2 electrons.
• \(n=2\): Can hold 8 electrons.
• \(n=3\): Can hold 18 electrons.

Recognizing the connection between the principal quantum number and electron capacity allows students to predict and understand atomic structures with ease and accuracy.