Ideal solution

CHAPTER 1: CO 2 CAPTURE

3. Theory of Solution

3.4 Ideal solution

An ideal solution is a solution with thermodynamic properties similar to those of a mixture of ideal gases. In an ideal solution, the interactions and entropies between molecules of the components does not differ from the interactions and entropies between the molecules of each component. Normally, this is a solution that each component obeys Raoult's Law. Raoult’s Law, established by French chemist Raoult (1887), states that for closely related liquids the partial vapor pressure (y_#) (Pascal) of each component in a solution is given by the product of the vapor pressure of the corresponding pure component (y_#^∗)!at the same temperature and its mole fraction ($_#) in the solvent:

P_f! = x_fP_f^∗!!!!!!_!

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! The chemical potential of ideal liquid solutions

For an ideal solution the chemical potential of all solution components in all concentrations and for the specific range of T and P follows the equation below:

n_f(l) = ! n_f^∗(ú)_!+ RTln LP

P_f^∗O

3128

where p_# and p_#^∗_! denote the chemical potential of component i and chemical potential of pure component i in the ideal solution, respectively. y_# is the vapor pressure of substance i and y_#^∗ is the vapor pressure of the pure component i at the same T. In all cases, the notation * means the pure

component. Since in ideal solution each component (i) obeys Raoult's Law, the chemical potential can be expressed as:

n_f(l) = ! n_f^∗_!(l) + RT_!!lnx_f! 3129

Mixtures that follow this rule throughout the whole composition range are called ideal solutions.

Many liquid solutions significantly deviate from the ideal behavior predicted by Raoul’s Law.

! Mixing properties of an ideal solution

From the definition of p_# in the equation 3-29 all the mixing properties can be easily obtained.

Based on

∆G_ìfä = G − G^∗ = ∑ x_{f f}(n_f− n_f^∗)!!!! 3130

The Gibbs free energy of mixing for ideal solutions follows immediately:

ü†_;('! = °D / *₍

!<*'₍!!!!!!!

3131

Since xi is the mole fraction of component i and in mixtures, all mole fractions are between 0 and 1, the logarithm of xi will be negative, in conclusion, ∆d_í#Å will have a negative amount as well.

According to the Second Law at constant P and T mixing is a spontaneous process. Based on equation (3-26), at T and P constant,!Δz_í#Å!= U^V¢ï£Y§!

V\ Z

[,W_Y!_!= 0 .

As we expect from the definition of an ideal solution, the formation of an ideal solution of its pure components at constant T and P, does not involve any changes in volume. The change in the molar entropy of the system is expressed by the following equation:

ü•_;('! = LMü†_;('!

MD O

F,*(

= −° / *₍

!<*'₍!

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Based on equation Δd_í#Å!= !Δa_í#Å!− xΔb_í#Å! and equations 3-31 and 3-32, the change of enthalpy for an ideal solution is zero. The zero-enthalpy change means that the ideal solution at constant T and P is not associated with any heat exchange. However, the reverse is not always true. If the Δa_{í#Å !}and Δz_{í#Å !} are zero the solution can be both ideal and non-ideal solution. Similarly, based on equation (3-126), and Δa_{í#Å !} = 0!and Δz_{í#Å !}= 0, the following relation can be obtained:

52

ü¶_;('! = ß!

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! Partial molar properties of an ideal solution Based on chemical potentials n_f = ! n_f^∗_!+ RT_!!lnx_f!, Sv_f= − ®^©™l

©i´

¨,k_l,!V__f= ®^©™l

©j´

i,k_l!and!H__f= n_f+ TSv_f all other thermodynamic partial molar properties for an ideal solution can be obtained straightforwardly.

•v₍ = ! •₍^∗_!− °_!!<*'_(!!!

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in equation 3-34 lnx_f! is negative in a mixture, therefore the partial molar entropy of a component of an ideal mixture is greater than the molar entropy of the pure substance at the same T and p.

9_₍ = ! ÆØ∞₍ ØF±

D,*(

= 9₍^∗_!!!!!!!

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≤_₍ = ∞₍^∗

!(+ D•₍^∗_!!_! = ! ≤₍^∗_!!!

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¶_₍= ≤₍^∗

!(+ F9₍^∗_!!_!= ! ¶₍^∗

!!!

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It is to be noted that in an ideal mixture the partial molar quantities of V__f, H__f and U__f are independent of the mixture composition and is equal to the molar quantities of pure component i at the same T and p as the mixture. These results are consistent with ΔV_ìfä!! = 0, ΔH_ìfä!!= 0 and ΔS_ìfä!!≠ 0. Note that ΔH_ìfä!!= 0!does not mean that there are no interactions, it means that the interactions in the mixture are the same as those in either pure substance.

! Ideal dilute solution

The ideal dilute solution is so dilute that the solute molecules interact mainly with solvent molecules. Many liquid solutions deviate from the ideal behavior predicted by Raoult's Law.

William Henry, an English chemist, found experimentally that for real solutions at low solute concentrations, despite the vapor pressure of the solute is proportional to the mole fraction,

contrary to Raoul’s law the constant of proportionality is not the same as the vapor pressure of the pure material. This relationship is defined as Henry’s Law and expressed by the equation below:

F_¥!!= '_¥!µ_¥!!!!_!! ³¹³⁸

where KB is an empirically determined constant with the units of pressure and XB is the mole fraction of the solute. B refers to the component B.

Henry's law constant can often be written in terms of the molarity and molality:

F₍ = µ_(,8∂₍! ³¹³⁹

F₍ = µ_(,;'₍! ³¹⁴⁰

where K_f,∏ and K_f,ì are related to K_f!.The subscripts c and m refer to the morality and molality, respectively. Ideal-dilute solutions are those for which the solute obeys Henry’s law and the solvent obeys Raoult’s Law. The reason that the solvent and solute are remarkably different in their behavior arises from the fact that in dilute solution the solvent molecules which are in large excess are surrounded by other solvent molecules so that they experience the environment very much like a pure liquid. (the solvent obeys Raoult’s Law). On the other hand, the solute which is in low concentration tends to be surrounded by solvent molecules and consequently, their environment and thermodynamic behavior are quite different from a pure one (The solute obeys Henry’s Law).

! Partial molar quantities in ideal-dilute solutions

The partial molar properties for the solvent and solute in ideal-dilute solutions are described as follows:

Solvent (A):

≤__π = ! ≤^∗_π! ³¹⁴¹

9__π = ! 9^∗π!!!!!!!!!!! ³¹⁴²

•v_π = ! •^∗_π− °<*!'_π ^3143!

Solute (B):

•v_∫ = ! •^∗_¥− °<*!'_¥ ^3144!

54 V__f = ª∂n_f^Ω

∂Pæ

i

!!!!!! ^3145!

Based on Sv_f= − ®^©™lø

©i´

j,kf and H__f= n_f^!+ TSv_f, partial enthalpy, and entropy will be obtained:

9__∫!= 9_^°_∫=9_^¡_∫! ³¹⁴⁶

≤__∫!= ≤_^°_∫=≤_^¡_∫! ^3147!

The partial molar volume and internal energy of the solute are constant in the ideal-dilute range and are equal to the values at infinite dilution (M_^¡_¬, (M = V, H)!). When the pressure is equal to the standard pressure p∘, the ( M_^¡_¬) quantities are equal to the standard values (M_^°_¬).

•v_∫ = ! •^∗_∫− °<*!'_∫ ^3148!

!

•v_∫! = ! •v^°_∫− °!<*'_∫!!!!!!!!! ^3149!