WEBVTT
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An ideal monatomic gas at a temperature of 542 kelvin expands adiabatically and reversibly to 4.00 times its volume.
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What is the gasβs final temperature?
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Considering this problem statement, one of the key things weβre told is that when the gas expands, it does so adiabatically, that is, without exchanging heat with the system.
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In the case of an expansion like this, we can say that the temperature of the gas multiplied by the volume raised to the power of πΎ minus one β weβll explain what πΎ is in a second β is equal to a constant.
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This means that if we have an initial state, that is, before the expansion, and a final state, after the expansion, we can write that π sub π times π sub π to the πΎ minus one is equal to π sub π times π sub π to the πΎ minus one.
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Now about this factor πΎ, this factor πΎ is equal to a ratio: the specific heat of the gas at a constant pressure, π, to the specific heat of the gas at a constant volume, π.
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The fact that our ideal gas is monatomic in this example implies a particular value for πΎ.
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It implies that the ratio πΆ sub π to πΆ sub π is equal to five-thirds.
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Going back to our expression for temperature then, we want to solve for the final temperature of the system, π sub π.
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Rearranging, thatβs equal to π sub π times the ratio π sub π over π sub π, all raised to the πΎ minus one power.
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In the problem statement, weβre told that π sub π is 542 kelvin and that π sub π is four times greater than π sub π.
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In other words, π sub π over π sub π is one-quarter.
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Since we also know the value for πΎ, weβre ready to plug in and solve for π sub π.
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Five-thirds minus one is two-thirds.
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And when we plug this expression into our calculator, to three significant figures, we find that π sub π is 215 kelvin.
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Thatβs the final temperature of the gas.