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Encyclopedia > Fugacity

Fugacity is a measure of the tendency of a substance to prefer one phase (liquid, solid, gas) over another. At a fixed temperature and pressure, water (for example) will have a different calculated fugacity for each phase. The phase with the lowest fugacity will thermodynamically be the most favorable: the one that minimizes Gibbs free energy. Fugacity, therefore, is a useful engineering tool for predicting the phase state of multi-component mixtures at various temperatures and pressures without doing the actual lab test. And besides predicting the preferred solid, liquid, or vapor phase, fugacity also applies to solid-solution equilibria. Shortcut: WP:CU Marking articles for cleanup This page is undergoing a transition to an easier-to-maintain format. ... This Manual of Style has the simple purpose of making things easy to read by following a consistent format — it is a style guide. ... Fig. ... The use of water pressure - the Captain Cook Memorial Jet in Lake Burley Griffin, Canberra. ... In thermodynamics, the Gibbs free energy is a thermodynamic potential which measures the useful work obtainable from a closed thermodynamic system at a constant temperature and pressure. ...


Fugacity is also a useful way to explain otherwise obscure behaviors of substances. For example: Most people are at a loss to describe why liquid water in a dish on a typical day eventually dries up. In the absence of heat energy to convert the water to a vapor, what causes water to spontaneously change phases from liquid to gas? Fugacity provides an answer: When the temperature and pressure is such that the relative humidity in the room is less than 100%, the calculated fugacity of water vapor will be lower than either liquid or solid water. Therefore, liquid water has a lower Gibbs free energy in the vapor phase and will evaporate. Fundamentally, water in a puddle is more ordered, or has less entropy than water dispersed all over the room as individual molecules. As dictated by the second law of thermodynamics all processes are headed towards maximum disorder, or increasing entropy, and water in a puddle is no exception. This article does not cite its references or sources. ... In thermodynamics, the Gibbs free energy is a thermodynamic potential which measures the useful work obtainable from a closed thermodynamic system at a constant temperature and pressure. ... Ice melting - classic example of entropy increasing[1] described in 1862 by Rudolf Clausius as an increase in the disgregation of the molecules of the body of ice. ... In science, a molecule is the smallest particle of a pure chemical substance that still retains its chemical composition and properties. ... The second law of thermodynamics is a theorem in physics regarding the directional flow of heat in relation to work and which accounts for the phenomenon of irreversibility in thermodynamic systems. ... Ice melting - classic example of entropy increasing[1] described in 1862 by Rudolf Clausius as an increase in the disgregation of the molecules of the body of ice. ...

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Technical Detail

In thermodynamics, the fugacity is a state function of matter at fixed temperature. fugacity is the change in pressure requred to make a real gas behave like an ideal gas.The fugacity, which has units of pressure, represents the tendency of a fluid to escape or expand isothermally. For gases at low pressures where the ideal gas law is a good approximation, fugacity is nearly equal to pressure. The ratio phi = f/P , between fugacity f, and pressure P, is called the fugacity coefficient. For an ideal gas, phi = 1 ,. Thermodynamics (from the Greek thermos meaning heat and dynamics meaning power) is a branch of physics that studies the effects of changes in temperature, pressure, and volume on physical systems at the macroscopic scale by analyzing the collective motion of their particles using statistics. ... In thermodynamics, a state function (or state quantity) is a property of a system that depends only on the current state of the system, not on the way in which the system got to that state. ... In physics, matter is commonly defined as the substance of which physical objects are composed, not counting the contribution of various energy or force-fields, which are not usually considered to be matter per se (though they may contribute to the mass of objects). ... Fig. ... The use of water pressure - the Captain Cook Memorial Jet in Lake Burley Griffin, Canberra. ... A fluid is defined as a substance that continually deforms (flows) under an applied shear stress regardless of the magnitude of the applied stress. ... An isothermal process is a thermodynamic process in which the temperature of the system stays constant; ΔT = 0. ... A gas is one of the four major phases of matter (after solid and liquid, and followed by plasma, that subsequently appear as a solid material is subjected to increasingly higher temperatures. ... Isotherms of an ideal gas The ideal gas law is the equation of state of an ideal gas. ... An ideal gas or perfect gas is a hypothetical gas consisting of identical particles of negligible volume, with no intermolecular forces. ...


For a given temperature T,, the fugacity f, satisfies the following differential relation:

d ln {f over f_0} = {dG over RT} = {{tilde V dP} over RT} ,

where G, is the Gibbs free energy, R, is the gas constant, tilde V, is the fluid's molar volume, and f_0, is a reference fugacity which is generally taken as that of an ideal gas at 1 bar. For an ideal gas, when f = P, this equation reduces to the ideal gas law. In thermodynamics, the Gibbs free energy is a thermodynamic potential which measures the useful work obtainable from a closed thermodynamic system at a constant temperature and pressure. ... The gas constant (also known as the universal or ideal gas constant, usually denoted by symbol R) is a physical constant used in equations of state to relate various groups of state functions to one another. ... In chemistry, the molar volume of a substance is the ratio of the volume of a sample of that substance to the amount of substance (usually in mole) in the sample. ... Isotherms of an ideal gas The ideal gas law is the equation of state of an ideal gas. ...


Thus, for any two mutually-isothermal physical states, represented by subscripts 1 and 2, the ratio of the two fugacities is as follows: An isothermal process is a thermodynamic process in which the temperature of the system stays constant; ΔT = 0. ...

f_2 / f_1 = exp left ({1 over RT} int_{G_1}^{G_2} dG right) = exp left ({1 over RT} int_{P_1}^{P_2} tilde V,dP right) ,

The concept of fugacity was introduced by American chemist Gilbert N. Lewis in his paper "The osmotic pressure of concentrated solutions, and the laws of the perfect solution," J. Am. Chem. Soc. 30, 668-683 (1908). Lewis in the Berkeley Lab Gilbert Newton Lewis (October 23, 1875-March 23, 1946) was a famous American physical chemist. ...


Fugacity and Chemical Potential

For ideal gases, we have the relation An ideal gas or perfect gas is a hypothetical gas consisting of identical particles of negligible volume, with no intermolecular forces. ...


dG = - SdT + VdP ,


for Gibbs free energy In thermodynamics, the Gibbs free energy is a thermodynamic potential which measures the useful work obtainable from a closed thermodynamic system at a constant temperature and pressure. ...


and


mu _i = left( {frac{{partial G}} {{partial n_i }}} right)_{T,P,n_{j ne i} }


for the chemical potential of the gas i In thermodynamics and chemistry, chemical potential, symbolized by μ, is a term introduced in 1876 by the American mathematical physicist (Willard Gibbs and his partner Lauren Berkley), which he defined as follows: Gibbs noted also that for the purposes of this definition, any chemical element or combination of elements in given...


If we choose to study only one pure ideal gas, we can divide both members of the equation by n and we’ll have


dmu = - bar SdT + bar VdP


where bar S and bar V are the molar entropy and molar volume for that gas Ice melting - classic example of entropy increasing[1] described in 1862 by Rudolf Clausius as an increase in the disgregation of the molecules of the body of ice. ... Volume is a quantification of how much space a certain region occupies. ...


We choose to study a process where T remains constant.


Therefore,


dmu = bar VdP


We can integrate this expression remembering the chemical potential is a function of T and P. We must also set a reference state. In this case, for an ideal gas the only reference state will be the pressure, and we set P = 1 bar.


int_{mu^circ }^mu {dmu } = int_{P^circ }^P {bar VdP}


Now, for the ideal gas bar V = frac{{RT}}{P}


mu - mu ^circ = int_{P^circ }^P {frac{{RT}} {P}dP} = RTln frac{P} {{P^circ }}


Reordering, we get


mu = mu ^circ + RTln frac{P} {{P^circ }}


Which gives the chemical potential for an ideal gas in an isothermal process, where the reference state is P=1 bar.


For a real gas, we cannot calculate int_{P^circ }^P {bar VdP} because we do not have a simple expression for a real gas’ molar volume. On the other hand, even if we did have one expression for it (we could use the Van der Waals equation, Redlich-Kwong or any other equation of state), it would depend on the substance being studied and would be therefore of a very limited usability. The van der Waals equation is an equation of state for a fluid composed of particles that have a non-zero size and a pairwise attractive inter-particle force (such as the van der Waals force. ... In physics and thermodynamics, an equation of state is a constitutive equation describing the state of matter under a given set of physical conditions. ...


We would like the expression for a real gas’ chemical potential to be similar to the one for an ideal gas.


We can define a magnitude, called fugacity, so that the chemical potential for a real gas becomes


mu = mu ^circ + RTln frac{f} {{f^circ }}


with a given reference state (discussed later).


We can see that for an ideal gas, it must be f = P


But for P to 0, every gas is an ideal gas. Therefore, fugacity must obey the limit equation In mathematics, the limit of a function is a fundamental concept in mathematical analysis. ...


mathop {lim }_{P to 0} frac{f} {P} = 1


We determine f by defining a function


Phi = frac{{Pbar V - RT}} {P}


We can obtain values for Φ experimentally easily by measuring V, T and P. (note that for an ideal gas, Φ = 0)


From the expression above we have


bar V = frac{{RT}} {P} + Phi


We can then write


int_{mu ^circ }^mu {dmu } = int_{P^circ }^P {bar VdP} = int_{P^circ }^P {frac{{RT}} {P}dP} + int_{P^circ }^P {Phi dP}


Where


mu = mu ^circ + RTln frac{P} {{P^circ }} + int_{P^circ }^P {Phi dP}


Since the expression for an ideal gas was chosen to be mu = mu ^circ + RTln frac{f} {{f^circ }},we must have


mu ^circ + RTln frac{f} {{f^circ }} = mu ^circ + RTln frac{P} {{P^circ }} + int_{P^circ }^P {Phi dP}


Rightarrow RTln frac{f} {{f^circ }} - RTln frac{P} {{P^circ }} = int_{P^circ }^P {Phi dP}


RTln frac{{fP^circ }} {{Pf^circ }} = int_{P^circ }^P {Phi dP}


Suppose we choose P to 0. Since mathop {lim }_{P to 0} f = P, we obtain


RTln frac{f} {P} = int_0^P {Phi dP}


The fugacity coefficient will then verify


ln phi = frac{1} {{RT}}int_0^P {Phi dP}


The integral can be evaluated via graphical integration if we measure experimentally values for Φ while varying P.


We can then find the fugacity coefficient of a gas at a given pressure P and calculate


f = phi P,


The reference state for the expression of a real gas’ chemical potential is taken to be “ideal gas, at P = 1 bar and work T”. Since in the reference state the gas is considered to be ideal (it is an hypothetical reference state), we can write that for the real gas


mu = mu ^circ + RTln frac{f} {{P^circ }}


Alternative Methods for calculating fugacity

If we suppose that Φ is constant between 0 and P (assuming it is possible to do this approximation), we have


frac{f} {P} = e^{frac{{Phi P}} {{RT}}}


Expanding in Taylor series about 0, As the degree of the Taylor series rises, it approaches the correct function. ...


frac{f} {P} approx 1 + frac{{Phi P}} {{RT}} = 1 + frac{1} {{RT}}left( {frac{{Pbar V - RT}} {P}} right)P = 1 + frac{{Pbar V}} {{RT}} - 1 = frac{{Pbar V}} {{RT}}


Finally, we get


f approx frac{{P^2 bar V}} {{RT}}


This formula allows us to calculate quickly the fugacity of a real gas at P,T, given a value for V (which could be determined using any equation of state), if we suppose is constant between 0 and P. In physics and thermodynamics, an equation of state is a constitutive equation describing the state of matter under a given set of physical conditions. ...


We can also use generalized charts for gases in order to find the fugacity coefficient for a given reduced temperature. In thermodynamics, the reduced temperature of a fluid means the actual temperature, divided by its critical temperature. ...


Fugacity could be considered a “corrected pressure” for the real gas, but should never be used to replace pressure in equations of state (or any other equations for that matter). That is, it is false to write expressions such as


fV = nRT


Fugacity is strictly a tool, conveniently defined so that the chemical potential equation for a real gas turns out to be similar to the equation for an ideal gas.


External links

  • Lewis' 1908 paper

  Results from FactBites:
 
Fugacity - Wikipedia, the free encyclopedia (945 words)
Fugacity provides an answer: When the temperature and pressure is such that the relative humidity in the room is less than 100%, the calculated fugacity of water vapor will be lower than either liquid or solid water.
Fugacity could be considered a “corrected pressure” for the real gas, but should never be used to replace pressure in equations of state (or any other equations for that matter).
Fugacity is strictly a tool, conveniently defined so that the chemical potential equation for a real gas turns out to be similar to the equation for an ideal gas.
  More results at FactBites »

 
 

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