Video: Calculating the Angular Momentum of an Electron in the Hydrogen Atom

In the Bohr model of the atom, what is the magnitude of the angular momentum of an electron in a hydrogen atom for which 𝑛 = 2? Use a value of 1.05 × 10⁻³⁴ J⋅s for the reduced Planck constant.

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Video Transcript

In the Bohr model of the atom, what is the magnitude of the angular momentum of an electron in a hydrogen atom for which 𝑛 equals two? Use a value of 1.05 times 10 to the negative 34 joule seconds for the reduced Planck constant.

So in this question, we’re considering a hydrogen atom, and we’ve specifically been asked to use the Bohr model of the atom. We can recall that a hydrogen atom just has one proton in the nucleus and one electron and the Bohr model describes atoms as consisting of electrons making circular orbits around the nucleus. So we can visualize this hydrogen atom like this. Here’s the nucleus, and here’s an electron making a circular path around it.

Let’s also recall that the Bohr model actually only makes accurate predictions for single-electron systems, which is why we’re being asked about hydrogen atom in this question. We’re being asked to find the angular momentum of an electron for which 𝑛 equals two. Let’s recall that 𝑛 is the principal quantum number of an electron. And it describes the energy level that the electron occupies. So an electron in the lowest possible energy level would have 𝑛 equals one. And in the Bohr model, this would refer to the innermost orbit around the nucleus.

In this question, we’re told that our electron has 𝑛 equals two, which would mean, according to the Bohr model, our electron occupies an orbit that’s further away from the nucleus. The Bohr model gives us a simple way of calculating the angular momentum of an electron in a hydrogen atom as long as we know what its principal quantum number is. In other words, we can find its angular momentum if we know which energy level it’s in.

This is given by the equation 𝐿 equals 𝑛ℎ bar, where 𝐿 represents the angular momentum of an electron, 𝑛 represents the principal quantum number, and ℎ bar is the reduced Planck constant. And we’re told in the question that this constant takes a value of 1.05 times 10 to the power of negative 34 joule seconds. We should note that even though it’s more common to express angular momentum in units of kilograms meters squared per second, these units are actually equivalent to the units of the reduced Planck constant.

Since the principal quantum number 𝑛 is dimensionless, this means that the units on the left and the right of the equation are equivalent. Since we’re looking to calculate the angular momentum and angular momentum is already the subject of this equation, all we need to do is multiply the principal quantum number of our electron by the reduced Planck constant. This gives us an angular momentum of two times 1.05 times 10 to the negative 34 joule seconds, which gives us a value of 2.10 times 10 to the negative 34 joule seconds. And this is the final answer to the question. In the Bohr model of the atom, the magnitude of the angular momentum of an electron in a hydrogen atom for which 𝑛 equals two is 2.10 times 10 to the negative 34 joule seconds.

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