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In the Bohr model of the atom, what is the magnitude of the angular momentum of an electron in a hydrogen atom in the ground state?
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Use a value of 1.05 times 10 to the negative 34 joule seconds for the reduced Planck constant.
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We can start by recalling that the Bohr model is a simplified model of the atom that describes electrons as making circular orbits around atomic nuclei.
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The Bohr model is based on the idea that the angular momentum of electrons is quantized.
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This means that, according to the Bohr model, electrons can only have certain specific values of angular momentum.
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This quantization of angular momentum in the Bohr model leads to the prediction that electrons can only occupy certain specific orbits around the nucleus.
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So each of these orbits represents a possible state that an electron can exist in within that atom.
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Each of these states is denoted by a specific value of a number π, which is also called the principal quantum number.
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By convention, the state corresponding to the orbit closest to the nucleus is given π equals one.
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Then, the next orbit out is given π equals two.
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Then, the next furthest out has π equals three, and so on.
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Each of these states corresponds to a specific value of angular momentum as well as a specific value of electron energy and a specific orbital radius, which describes how far the electron is from the nucleus.
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The π equals one state is also known as the ground state.
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An electron in this state will have the lowest possible orbital radius, the lowest possible amount of energy, and the lowest possible angular momentum.
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And as we look at higher values of π, all of these quantities increase.
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Note that this question specifically asks us about the angular momentum of an electron in a hydrogen atom.
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This is because the Bohr model only really makes accurate predictions for atoms with a single electron.
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This means it works well for predicting the behavior of hydrogen atoms, since these only have one electron.
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Now, weβve talked about the fact that in the Bohr model, angular momentum of electrons is quantized.
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But this question is asking us to actually calculate the magnitude of the angular momentum of an electron.
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In order to do this, we need to use this equation that comes from the Bohr model.
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πΏ equals π times β bar, where πΏ is the angular momentum of an electron, π is the principal quantum number that describes the state of the electron, and β bar is a physical constant known as the reduced Planck constant, which weβve been given a value for in the question.
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This equation enables us to calculate the angular momentum of an electron with any given principal quantum number.
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All we need to do is multiply its principal quantum number, that is, its value of π, by the reduced Planck constant.
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Now, in this question, weβre being asked to calculate the magnitude of the angular momentum of an electron in the ground state of a hydrogen atom.
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Ground state just means π equals one.
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So all we need to do is substitute π equals one into this equation.
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Doing this, we have that the angular momentum πΏ is equal to one times β bar, which of course we can simplify to just πΏ equals β bar.
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And weβre told in the question that β bar has a value of 1.05 times 10 to the power of negative 34 joule seconds.
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So this is the final answer to the question.
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Because the Bohr model tells us that the angular momentum πΏ of an electron is just equal to its principal quantum number multiplied by β bar, and an electron in the ground state has π equal to one, then the angular momentum of an electron in the ground state is just equal to β bar.