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An electron in an Li²⁺ ion moves from the 𝑛 equals two energy level to the 𝑛 equals one energy level.
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Using the Bohr model, calculate, to four significant figures, the energy of the photon produced by this transition.
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Li²⁺ is lithium two plus.
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Lithium has an atomic number of three.
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So, the lithium nucleus contains three protons.
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We can take away the charge two plus from the number of protons to work out the number of electrons.
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In this case, we have a single electron.
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For this question, we don’t need to worry about the neutrons.
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In the Bohr model, the only thing that matters is the charges of the nucleus and the electron.
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In the Bohr model, the nucleus is surrounded by concentric shells.
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Electrons are said to occupy these shells at fixed restricted distances from the nucleus.
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Because of the way this system is sometimes drawn, the Bohr model is sometimes called the planetary model.
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However, electrons aren’t like planets.
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They’re more like waves than particles.
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And the shells are not rings; they are surfaces of spheres.
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Each shell is assigned a number, with the number one given to the shell closest to the nucleus.
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The lower the value of 𝑛, the more stable the electron will be if inside that shell.
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Each shell has an energy, but the zero point is placed where the electron is infinitely far from the nucleus.
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This is the equivalent of a shell number of infinity.
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And it’s the equivalent, in this case, of having a lithium three plus nucleus and a completely separated electron.
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At the other end of the spectrum, we have the inner shell, where 𝑛 equals one, where the energy is considered negative.
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When an electron moves from a high energy level to a low energy level, the difference in energy is released in the form of a photon.
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In this question, we’re being asked to work out the energy of the photon released when an electron moves from the 𝑛 equals two energy level to the 𝑛 equals one energy level.
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So, we’re going from the first excited state of lithium two plus to the ground state, releasing a photon of energy Eph.
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Before we go any further, we’re going to need to work out the energy of a shell based on its shell number.
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This is the equation from the Bohr model that tells you the energy of a shell based on the shell number.
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𝑍 is equal to the number of protons in the nucleus, while 𝑛 is equal to the shell number.
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But what about this term here?
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Well, the full expression for the energy of any given shell is quite a complex combination of many constants of nature.
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In this expression, we’re combining the Coulomb constant, the elementary charge, the electron mass, and the reduced Planck constant.
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But this is a constant times, a constant divided by another constant.
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It all resolves to a constant.
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And that constant has a value of about 13.6 electron volts.
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If you take the values of the constants in their SI units, you’ll get an answer in joules.
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And you can convert to electron volts by dividing the value in joules by the elementary charge.
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Thankfully, all that work has been done, so we have our single condensed constant term of negative 13.6057 electron volts.
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Now, all we need to do is work out the values for 𝐸 two and 𝐸 one and work out the difference and calculate the energy of the released Photon.
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The energy of our core shell is equal to minus 13.6057 electron volts multiplied by three squared over one squared.
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𝑍 is the number of protons in the lithium nucleus, which is three.
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And 𝑛 equals one.
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This evaluates to minus 122.451 electron volts.
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The value of our second energy level, 𝐸 two, is equal to negative 13.6057 electron volts multiplied by three squared divided by two squared, which gives us negative 30.6128 electron volts.
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Now, we can put those values into our diagram and work out the difference in energy.
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Our difference in energy is the final energy minus the initial energy, which is equal to negative 122.451 electron volts minus negative 30.6128 electron volts, giving us a change in energy of minus 91.8382 electron volts.
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So, this is the amount of energy that the electron has lost, becoming more stable.
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So, the energy of the photon is the negative of this energy, the exact opposite.
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Because energy can neither be created nor destroyed.
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Giving us an energy for the photon of positive 91.8382 electron volts.
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We can also express this value in joules, which is 1.47141 times 10 to the minus 17 joules.
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We convert from electron volts to joules by multiplying by 1.60218 times 10 to the minus 19 joules per electron volt.
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The question asks for our answer to be to four significant figures, which is 91.84 electron volts, or 1.471 times 10 to the minus 17 joules.
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So, we’ve worked out that when an electron moves from the 𝑛 equals two energy level to the 𝑛 equals one energy level, in the Bohr model, the energy of the photon produced is 1.471 times 10 to the minus 17 joules.