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| The energy of the [[ideal gas]] is given by (Hill Eq. 4-16) | | The energy of the [[ideal gas]] is given by (Hill Eq. 4-16) |
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| :<math>E = -T^2 \left. \frac{\partial (A/T)}{\partial T} \right\vert_{V,N} = kT^2 \left. \frac{\partial \ln Q}{\partial T} \right\vert_{V,N}= NkT^2 \frac{d \ln T^{3/2}}{dT} = \frac{3}{2} NkT \equiv \frac{3}{2} RT </math> | | :<math>E = kT^2 \left. \frac{\partial \ln Q}{\partial T} \right\vert_{V,N}= NkT^2 \frac{d \ln T^{3/2}}{dT} = \frac{3}{2} NkT</math> |
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| where <math>R</math> is the [[molar gas constant]].
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| This energy is all ''kinetic energy'', <math>1/2 kT</math> per [[degree of freedom]], by [[equipartition]]. This is because there are no intermolecular forces, thus no potential energy. This result is valid only for a monoatomic ideal gas. The general expression would be
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| :<math>E = \frac{n}{2} NkT = \frac{n}{2} RT, </math>
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| where <math>n</math> is the number of degrees of freedom. This number is 3 for atoms; if would be 6 in principle for diatomic molecules, but in normal conditions 5 is a very good approximation since vibrations are "frozen" (as explained in the entry about [[degree of freedom | degrees of freedom]].)
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| | This energy is all ''kinetic energy'', <math>1/2.kT</math> per degree of freedom. This is because there are no intermolecular forces, thus no potential energy. |
| ==References== | | ==References== |
| #Terrell L. Hill "An Introduction to Statistical Thermodynamics" 2nd Ed. Dover (1962) | | #Terrell L. Hill "An Introduction to Statistical Thermodynamics" 2nd Ed. Dover (1962) |
| [[category: ideal gas]] | | [[category: ideal gas]] |