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and you have an entropy that's volume
dependent and the minimum volume is
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dependent and the minimum volume is
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dependent and the minimum volume is
obtained by minimizing this with respect
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obtained by minimizing this with respect
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obtained by minimizing this with respect
to uh the volume okay there's another
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to uh the volume okay there's another
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to uh the volume okay there's another
way you can show where thermal expansion
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way you can show where thermal expansion
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way you can show where thermal expansion
comes from and here I have to brush up
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comes from and here I have to brush up
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comes from and here I have to brush up
my own Thermo again um but essentially
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my own Thermo again um but essentially
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my own Thermo again um but essentially
you can show that it comes from the
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you can show that it comes from the
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you can show that it comes from the
volume dependence that you can get it
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volume dependence that you can get it
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volume dependence that you can get it
from only this term from the volume
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from only this term from the volume
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from only this term from the volume
dependence of the entropy because the
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dependence of the entropy because the
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dependence of the entropy because the
volume dependence
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volume dependence
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volume dependence
of the entropy
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of the entropy
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of the entropy
here you can use all this beautiful