Virial equation of state: Difference between revisions

Revision as of 11:15, 12 July 2007

The virial equation of state is used to describe the behavior of diluted gases. It is usually written as an expansion of the compresiblity factor, $Z$ , in terms of either the density or the pressure. Such an expansion was first introduced by Kammerlingh Onnes. In the first case:

{\frac {pV}{Nk_{B}T}}=Z=1+\sum _{k=2}^{\infty }B_{k}(T)\rho ^{k-1}

.

where

$p$ is the pressure

$V$ is the volume

$N$ is the number of molecules

$\rho \equiv {\frac {N}{V}}$ is the (number) density

$B_{k}\left(T\right)$ is called the k-th virial coefficient

Virial coefficients

The second virial coefficient represents the initial departure from ideal-gas behavior

B_{2}(T)={\frac {N_{0}}{2V}}\int ....\int (1-e^{-\Phi /k_{B}T})~d\tau _{1}d\tau _{2}

where $N_{0}$ is Avogadros number and $d\tau _{1}$ and $d\tau _{2}$ are volume elements of two different molecules in configuration space.

One can write the third virial coefficient as

B_{3}(T)=-{\frac {1}{3V}}\int \int \int f_{12}f_{13}f_{23}dr_{1}dr_{2}dr_{3}

where f is the Mayer f-function.

References

H. Kammerlingh Onnes "", Communications from the Physical Laboratory Leiden 71 (1901)
James A Beattie and Walter H Stockmayer "Equations of state", Reports on Progress in Physics 7 pp. 195-229 (1940)

@@ Line 17: / Line 17: @@
 *<math> B_k\left( T \right) </math> is called the k-th virial coefficient
 ==Virial coefficients==
-The second virial coefficient represents the initial departure from ideal-gas behavior
+The [[second virial coefficient]] represents the initial departure from ideal-gas behavior
-<math>B_{2}(T)= \frac{N_0}{2V} \int .... \int (1-e^{-u/kT}) ~d\tau_1 d\tau_2
+:<math>B_{2}(T)= \frac{N_0}{2V} \int .... \int (1-e^{-\Phi/k_BT}) ~d\tau_1 d\tau_2</math>
-</math>
 where <math>N_0</math> is [[Avogadro constant | Avogadros number]] and <math>d\tau_1</math> and <math>d\tau_2</math> are volume elements of two different molecules
-in configuration space. The integration is to be performed over all available phase-space; that is,
+in configuration space.
-over the volume of the containing vessel.
-For the special case where the molecules posses spherical symmetry, so that <math>u</math> depends not on
-orientation, but only on the separation <math>r</math> of a pair of molecules, the equation can be simplified to
-:<math>B_{2}(T)= - \frac{1}{2} \int_0^\infty \left(\langle \exp\left(-\frac{u(r)}{k_BT}\right)\rangle -1 \right) 4 \pi r^2 dr</math>
+One can write the third virial coefficient as
-Using the  [[Mayer f-function]]
+:<math>B_{3}(T)= - \frac{1}{3V} \int \int \int f_{12} f_{13} f_{23}  dr_1 dr_2 dr_3</math>
-:<math>f_{ij}=f(r_{ij})= \exp\left(-\frac{u(r)}{k_BT}\right) -1 </math>
+where ''f'' is the [[Mayer f-function]].
-one can write the third virial coefficient more compactly as
-:<math>B_{3}(T)= - \frac{1}{3V} \int \int \int f_{12} f_{13} f_{23}  dr_1 dr_2 dr_3
-</math>
 ==References==
 # H. Kammerlingh Onnes "", Communications from the Physical Laboratory Leiden '''71''' (1901)
-#[http://dx.doi.org/10.1088/0034-4885/7/1/312 James A Beattie and Walter H Stockmayer "Equations of state",Reports on Progress in Physics '''7''' pp. 195-229 (1940)]
+#[http://dx.doi.org/10.1088/0034-4885/7/1/312 James A Beattie and Walter H Stockmayer "Equations of state", Reports on Progress in Physics '''7''' pp. 195-229 (1940)]
 [[category:equations of state]]

Virial equation of state: Difference between revisions

Revision as of 11:15, 12 July 2007

Virial coefficients

References

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