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If an electron in a hydrogen atom is at a distance of 1.32 nanometers from the nucleus, what energy level is it in?
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Use a value of 5.29 times 10 to the negative 11 meters for the Bohr radius.
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A neutral hydrogen atom has one proton in its nucleus and one orbiting electron.
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According to the Bohr model of the atom, this electron moves around the nucleus in a circular orbit.
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As such, it has some orbital radius, and that radius is given by the following general equation.
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The orbital radius of an electron in the πth energy level of a hydrogen atom, π sub π, is equal to whatβs called the Bohr radius multiplied by π squared.
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The Bohr radius is the orbital radius of an electron in a hydrogen atom if itβs in the ground energy state.
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That state has a principal quantum number value of one.
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And if we substitute one in for π in this equation, we see that indeed the orbital radius of an electron in the first energy level in a hydrogen atom, that is, the ground state, is equal to the Bohr radius.
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In our case, weβre given a value for π sub π, 1.32 nanometers.
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By our equation, We know that this equals the Bohr radius π sub zero times π squared.
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Here, we want to solve for the principal quantum number π corresponding to the energy level of our electron.
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If we divide both sides of this equation by the Bohr radius π sub zero, canceling that factor on the right, and if we then take the square root of both sides so that the square root of π squared is equal simply to π, we find that the energy level π is equal to the square root of π sub π divided by π sub zero.
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As weβve seen, our electronβs orbital radius is 1.32 nanometers.
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And the Bohr radius π sub zero is 5.29 times 10 to the negative 11 meters.
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To move ahead with this calculation, letβs convert the units of our numerator from nanometers into meters.
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One nanometer, we recall, is equal to one one billionth of a meter.
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And therefore 1.32 nanometers equals 1.32 times 10 to the negative nine meters.
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Notice that now the units in our expression cancel out entirely so that π will be unitless.
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When we calculate π to the nearest whole number, itβs equal to five.
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This is the energy level of our hydrogen atom electron.