Virial equation of state: Difference between revisions

Revision as of 10:15, 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, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Z } , in terms of either the density or the pressure. Such an expansion was first introduced by Kammerlingh Onnes. In the first case:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{p V}{N k_B T } = Z = 1 + \sum_{k=2}^{\infty} B_k(T) \rho^{k-1}} .

where

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p } is the pressure

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V } is the volume

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N } is the number of molecules

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \rho \equiv \frac{N}{V} } is the (number) density

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B_{2}(T)= \frac{N_0}{2V} \int .... \int (1-e^{-\Phi/k_BT}) ~d\tau_1 d\tau_2}

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N_0} is Avogadros number and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d\tau_1} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d\tau_2} are volume elements of two different molecules in configuration space.

One can write the third virial coefficient as

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 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 10:15, 12 July 2007

Virial coefficients

References

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