Difference between revisions of "Ideal gas: Energy"

From SklogWiki
Jump to: navigation, search
m (Added R to equation.)
(Expression for more degrees of freedom)
 
Line 4: Line 4:
  
 
where <math>R</math> is the [[molar gas constant]].
 
where <math>R</math> is the [[molar gas constant]].
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 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
 +
:<math>E =  \frac{n}{2} NkT  = \frac{n}{2} RT, </math>
 +
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]].)
 +
 
 +
 
 
==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]]

Latest revision as of 16:44, 12 December 2008

The energy of the ideal gas is given by (Hill Eq. 4-16)

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

where R is the molar gas constant. This energy is all kinetic energy, 1/2 kT 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

E =   \frac{n}{2} NkT  = \frac{n}{2} RT,

where n 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 degrees of freedom.)


References[edit]

  1. Terrell L. Hill "An Introduction to Statistical Thermodynamics" 2nd Ed. Dover (1962)