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Which of the following graphs best describes the relationship between the amount of a gas, in moles, versus its temperature at constant pressure and volume?
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And for our options, we’ve been given five plots with different slopes and curves.
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For this question, we need to figure out what kind of relationship exists between the amount of gas, which is proportional to the number of molecules of gas and the temperature of the gas.
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Avogadro’s law tells us, all else being constant, the volume of a gas is proportional to the amount of gas.
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But we’ve been told that volume is constant.
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So we can’t apply Avogadro’s law.
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Gay-Lussac’s law tells us that pressure is proportional to temperature if everything else is constant.
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But we’ve been told that pressure is also constant.
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And lastly, Boyle’s law gives us the relationship between the pressure and volume of an ideal gas, assuming everything else is constant.
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But none of these laws directly address the relationship between the amount of gas and the temperature.
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For that, we’re going to need the ideal gas law, which takes the relationship between volume and amount from Avogadro’s law, the relationship between pressure and volume from Boyle’s law, the relationship between pressure and temperature from Gay-Lussac’s law and puts in the gas constant so that we have the correct numerical relationship.
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Now, just a word of warning, this is not how you should derive the ideal gas law.
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But it is quite elegant that the ideal gas law preserves all the relationships from the other gas laws.
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With the ideal gas law, we have the amount of gas and the temperature of that gas, both on the right-hand side.
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So let’s rearrange the equation so that we have 𝑛 in terms of 𝑇.
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So we flip it around and divide both sides by 𝑅𝑇, giving us 𝑛 is equal to 𝑃 multiplied by 𝑉 divided by 𝑅𝑇.
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The question stipulates that pressure and volume are constant.
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Therefore, pressure times volume divided by the gas constant is also a constant.
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Meaning we can express the relationship between amount and temperature of a gas as 𝑛 equals some constant 𝑘 divided by 𝑇.
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Now, we can start drawing the graph.
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The first thing we do is put amount on the 𝑦-axis and temperature on the 𝑥-axis because the question wants us to visualize amount versus temperature.
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We also join the axes at the origin zero, zero.
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For this question, we’re only trying to find the graph with the right shape.
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So we can assume a value for 𝑘 of one to make the mathematics easier.
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In reality, the value for 𝑘 is very unlikely to be one.
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But for this case, it doesn’t matter.
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Now, we can start plugging in some values.
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And we can see what shape it produces.
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If you plug in zero for the value of 𝑇 into your calculator, it’ll return an error because dividing by zero is not a valid function.
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But if we plug in a small value like 0.001, we’ll get quite a large value for 𝑛.
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This looks quite tricky to plot.
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So maybe we’re going about this the wrong way.
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Instead of starting from zero, let’s start from one.
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If we plug in temperature equal to one kelvin, 𝑛 is equal to one.
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So there’s our first point.
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If we plug in 𝑇 equal to two, 𝑛 is equal to a half.
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But if we plug in 𝑇 is equal to 0.5, 𝑛 is equal to two.
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Even though we’ve only got three points, the graph is already looking familiar.
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And we only need to plug in a couple of more values to be sure that it best resembles C.
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After drawing a line of best fit, we can clearly see the resemblance.
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This shape is characteristic of variables which are inversely proportional.
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Increasing the value of 𝑥 decreases the value of 𝑦 and vice versa.
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We can see from the ideal gas law that if everything else is constant, the amount of gas is inversely proportional to the temperature, just so long as we’re using kelvin.
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Although it should be said that this is quite an unusual relationship to study.
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Generally speaking, heating something up won’t change the amount.
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And changing the amount won’t directly change the temperature.
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So the data would have to be gathered from multiple experiments, where we use different amounts of gas in a fixed volume container and then change the temperature until the pressure hits a preset value.
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However it’s determined either from the ideal gas law or from experiment, we know that the shape of the graph best resembles C.
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So C is the graph that best resembles the relationship between amount of gas and its temperature at constant pressure and volume.