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Van der Waals equation of state

2009-02-19T13:19:50Z

Jfcastillo: small changes for consistency

The '''van der Waals equation of state''', developed by [[ Johannes Diderik van der Waals]], takes into account two features that are absent in the [[Equation of State: Ideal Gas | ideal gas]] equation of state; the parameter <math> b </math> introduces somehow the repulsive behavior between pairs of molecules at short distances,
it represents the minimum molar volume of the system, whereas <math> a </math> measures the attractive interactions between the molecules. The van der Waals equation of state leads to a liquid-vapor equilibrium at low temperatures, with the corresponding critical point.
==Equation of state==
The van der Waals equation of state can be written as

:<math> \left. p = \frac{ n R T}{V - n b } - a \left( \frac{ n}{V} \right)^2 \right. </math>.

where:
* <math> p </math> is the [[pressure]],
* <math> V </math> is the volume,
* <math> n </math> is the number of moles,
* <math> T </math> is the absolute [[temperature]],
* <math> R </math> is the [[molar gas constant]]; <math> R = N_A k_B </math>, with <math> N_A </math> being the [[Avogadro constant]] and <math>k_B</math> being the [[Boltzmann constant]].

:<math>a= \frac{27}{64}\frac{R^2T_c^2}{P_c}</math>

:<math>b= \frac{RT_c}{8P_c}</math>
==Critical point==
The [[Critical points |critical point]] for the van der Waals equation of state can be found at
:<math>T_c= \frac{8a}{27bR}</math>

:<math>p_c=\frac{a}{27b^2}</math>
and at
:<math>\left.V_c\right.=3b</math>.
==Dimensionless formulation==
If one takes the following reduced quantities

:<math>\tilde{p} = \frac{p}{p_c};~ \tilde{V} = \frac{V}{V_c}; ~\tilde{t} = \frac{T}{T_c};</math>

one arrives at

:<math>\tilde{p} = \frac{8\tilde{t}}{3\tilde{V} -1} -\frac{3}{\tilde{V}^2}</math>

The following image is a plot of the isotherms <math>T/T_c</math> = 0.85, 0.90, 0.95, 1.0 and 1.05 (from bottom to top) for the van der Waals equation of state:
[[Image:vdW_isotherms.png|center|Plot of the isotherms T/T_c = 0.85, 0.90, 0.95, 1.0 and 1.05 for the van der Waals equation of state]]
==Maxwell's equal area construction==
==Interesting reading==
*[http://nobelprize.org/nobel_prizes/physics/laureates/1910/waals-lecture.pdf Johannes Diderik van der Waals "The Equation of State for Gases and Liquids", Nobel Lecture, December 12, 1910]
*Luis Gonzalez MacDowell and Peter Virnau "El integrante lazo de van der Waals", Anales de la Real Sociedad Española de Química '''101''' #1 pp. 19-30 (2005)
==References==
*J. D. van der Waals "Over de Continuiteit van den Gas- en Vloeistoftoestand", doctoral thesis, Leiden, A,W, Sijthoff (1873).
English translation:
*[http://store.doverpublications.com/0486495930.html J. D. van der Waals "On the Continuity of the Gaseous and Liquid States", Dover Publications ISBN: 0486495930]
[[Category: equations of state]]

Van der Waals equation of state

2009-02-19T13:16:32Z

Jfcastillo: some typos corrected

The '''van der Waals equation of state''', developed by [[ Johannes Diderik van der Waals]], takes into account two features that are absent in the [[Equation of State: Ideal Gas | ideal gas]] equation of state; the parameter <math> b </math> introduces somehow the repulsive behavior between pairs of molecules at short distances,
it represents the minimum molar volume of the system, whereas <math> a </math> measures the attractive interactions between the molecules. The van der Waals equation of state leads to a liquid-vapor equilibrium at low temperatures, with the corresponding critical point.
==Equation of state==
The van der Waals equation of state can be written as

:<math> \left. p = \frac{ n R T}{V - n b } - a \left( \frac{ n}{V} \right)^2 \right. </math>.

where:
* <math> p </math> is the [[pressure]],
* <math> V </math> is the volume,
* <math> n </math> is the number of moles,
* <math> T </math> is the absolute [[temperature]],
* <math> R </math> is the [[molar gas constant]]; <math> R = N_A k_B </math>, with <math> N_A </math> being the [[Avogadro constant]] and <math>k_B</math> being the [[Boltzmann constant]].

:<math>a= \frac{27}{64}\frac{R^2T_c^2}{P_c}</math>

:<math>b= \frac{RT_c}{8P_c}</math>
==Critical point==
The [[Critical points |critical point]] for the van der Waals equation of state can be found at
:<math>T_c= \frac{8a}{27bR}</math>

:<math>p_c=\frac{a}{27b^2}</math>
and at
:<math>\left.V_c\right.=3b</math>.
==Dimensionless formulation==
If one takes the following quantities

:<math>\tilde{p} = \frac{p}{p_c};~ \tilde{v} = \frac{v}{v_c}; ~\tilde{t} = \frac{T}{T_c};</math>

one arrives at

:<math>\tilde{p} = \frac{8\tilde{t}}{3\tilde{v} -1} -\frac{3}{\tilde{v}^2}</math>

The following image is a plot of the isotherms <math>T/T_c</math> = 0.85, 0.90, 0.95, 1.0 and 1.05 (from bottom to top) for the van der Waals equation of state:
[[Image:vdW_isotherms.png|center|Plot of the isotherms T/T_c = 0.85, 0.90, 0.95, 1.0 and 1.05 for the van der Waals equation of state]]
==Maxwell's equal area construction==
==Interesting reading==
*[http://nobelprize.org/nobel_prizes/physics/laureates/1910/waals-lecture.pdf Johannes Diderik van der Waals "The Equation of State for Gases and Liquids", Nobel Lecture, December 12, 1910]
*Luis Gonzalez MacDowell and Peter Virnau "El integrante lazo de van der Waals", Anales de la Real Sociedad Española de Química '''101''' #1 pp. 19-30 (2005)
==References==
*J. D. van der Waals "Over de Continuiteit van den Gas- en Vloeistoftoestand", doctoral thesis, Leiden, A,W, Sijthoff (1873).
English translation:
*[http://store.doverpublications.com/0486495930.html J. D. van der Waals "On the Continuity of the Gaseous and Liquid States", Dover Publications ISBN: 0486495930]
[[Category: equations of state]]

Ideal solution

2009-02-19T12:11:32Z

Jfcastillo:

An ''' ideal solution''' is defined as a solution that obeys [[ Raoult's law]] for all concentrations.

==References==

[[category: classical thermodynamics]]

Ideal solution

2009-02-19T12:10:41Z

Jfcastillo:

An ''' ideal solution''' is defined as a solution that obeys [[ Raoult's law]] for all concentrations.

Ideal solution

2009-02-19T12:10:16Z

Jfcastillo: New page: An ideal solution is defined as an solution that obeys Raoult's law for all concentrations.

An ideal solution is defined as an solution that obeys [[ Raoult's law]] for all concentrations.

Dalton's law

2009-02-19T11:52:19Z

Jfcastillo:

Dalton's law stablishes that the partial pressure of a gas <math>i</math> in a ideal mixture of gases at pressure <math>P</math> is:
:<math> P_i = X_i \, P </math>
where <math>X_i</math> is the [[molar fraction]] of a gas <math>i</math>.

==References==

[[category: classical thermodynamics]]

Dalton's law

2009-02-19T11:46:06Z

Jfcastillo:

Dalton's law

2009-02-19T11:45:23Z

Jfcastillo:

Dalton's law stablishes that the partial pressure of a gas <math>i</math> in a ideal mixture of gases at pressure <math>P</math> is:
:<math> P_i = X_i P </math>
where <math>X_i</math> is the [[molar fraction]] of a gas <math>i</math>.

Dalton's law

2009-02-19T11:43:08Z

Jfcastillo:

Dalton's law stablishes that the partial pressure of a gas <math>i</math> in a ideal mixture of gases at pressure P is:
:<math> P_i = X_i P </math>

Dalton's law

2009-02-19T11:42:39Z

Jfcastillo: New page: Dalton's law stablishes that the partial pressure of a gas <math>i</math> in a ideal mixture of gases is: :<math> P_i = X_i P </math>

Dalton's law stablishes that the partial pressure of a gas <math>i</math> in a ideal mixture of gases is:
:<math> P_i = X_i P </math>

Molar fraction

2009-02-19T11:38:42Z

Jfcastillo:

The molar fraction of a component <math>i</math> in a mixture is defined as:
:<math>X_i = \frac{n_i}{\sum_{j=1}^{N} n_j} </math>
where <math> n_i </math> is the number of moles of component <math>i</math>.

Molar fraction

2009-02-19T11:36:51Z

Jfcastillo:

The molar fraction of a component i in a mixture is defined as:
:<math>X_i = \frac{n_i}{\sum_{j=1}^{N} n_j} </math>
where <math> n_i </math> is the number of moles of component i.

Raoult's law

2009-02-19T11:34:36Z

Jfcastillo:

'''Raoult's law''' states that the [[vapour pressure]] of an [[ideal solution]] of N components is:
:<math>P_v = \sum_{i=1}^{N} X_i P^*_{v,i} </math>

where <math>X_i</math> is the [[molar fraction]] of component <math>i</math>, and <math>P^*_{v,i}</math> is the vapour pressure of pure <math>i</math>.
More generally, Raoult's law describes the [[partial pressure]] of component <math>A</math> in the vapour coexisting with a liquid mixture as:

:<math> P_A = X_A P^*_{v,A} </math>.

This law is obeyed for all components of an ideal solution, and is also obeyed for the solvent of an [[ideal dilute solution]]. The solute's partial pressure of such solutions then obey [[ Henry's law]]. Ideal dilute solutions describe the limiting behaviour of a mixture of infinite dilution. Therefore, all solutions in the limit of infinite dilution obey Raoult's law, i.e.:

:<math> \lim_{X_A \rightarrow 1} P_A = X_A P^*_{v,A} </math>.
==References==

[[category: classical thermodynamics]]

Molar fraction

2009-02-19T11:32:55Z

Jfcastillo: New page: The molar fraction of a component i in a mixture is defined as: :<math>X_i = \frac{n_i}{\sum_{j=1}^{N} n_j} </math>

The molar fraction of a component i in a mixture is defined as:
:<math>X_i = \frac{n_i}{\sum_{j=1}^{N} n_j} </math>

Raoult's law

2009-02-18T17:21:59Z

Jfcastillo: New page: '''Raoult's law''' states that the vapour pressure of an ideal solution of two components :<math>P_v = X_A P^*_{v,A} + X_B P^*_{v,B}</math> category: classical thermodynamics

'''Raoult's law''' states that the [[vapour pressure]] of an [[ideal solution]] of two components
:<math>P_v = X_A P^*_{v,A} + X_B P^*_{v,B}</math>
[[category: classical thermodynamics]]