Question

What are the largest and smallest possible values for the angular momentum \(L\) of an electron in the \(n=5\) shell?

Short answer

Answer: The smallest possible value for the angular momentum of an electron in the n=5 shell is 0, and the largest possible value is $$\sqrt{20}\hbar$$.

Step-by-step solution

Identify the quantum numbers for the n=5 shell

For an electron in the n=5 shell, we have to find the values of the angular momentum quantum number l, which can range from 0 to n-1. In this case, the possible values of l are: 0, 1, 2, 3, and 4.

Calculate the smallest angular momentum

The smallest value for l is 0. We can use the formula for angular momentum to calculate the smallest possible value for L. $$L = \sqrt{l(l+1)}\hbar = \sqrt{0(0+1)}\hbar = 0.$$ So, the smallest possible value for the angular momentum of an electron in the n=5 shell is 0.

Calculate the largest angular momentum

Next, we need to calculate the largest possible value for L. Since the largest value for l is 4, we can use the formula for angular momentum as follows: $$L = \sqrt{l(l+1)}\hbar = \sqrt{4(4+1)}\hbar = \sqrt{20}\hbar.$$ Thus, the largest possible value for the angular momentum of an electron in the n=5 shell is $$\sqrt{20}\hbar$$.

State the largest and smallest angular momentum values

The smallest possible value for the angular momentum of an electron in the n=5 shell is 0, and the largest possible value is $$\sqrt{20}\hbar$$.