معامل فاعلية

معامل الفاعلية عدد لا بعدي في الديناميكا الحرارية يُستخدم ليُعلّل انحرافات عن السلوك «المثالي» لمزيج من المواد الكيماوية. ويتصل مفهوم «معامل الفاعلية» بمفهوم الفاعلية الكيميائية ذاتها بطبيعة الحال.

في كلّ مزيج مثالي ، تتطابق التآثرات بين كلّ نوعان كيميائيان - أيّ أنّ سخانة الخلط صفرًا - ولذلك يُمكن التعبير عن خصائص المزيج مباشرةً بدالة منطلقها تركيز كلّ مادة موجودة أو ضغطها الجزئي - مثلًا قانون راؤول. ويُعلّل الانحراف عن المثالية بتعديل التركيز باستخدام «معامل الفاعلية».

ويوجد معامل شبيه يُستخدم لتعديل تعبيرات تتعلق بالغازات بطريقة شبيهة اسمه « معامل الافتلات ».

الثرموديناميك

كمون ( μ_B ) أي مادة (B) تتواجد في مزيج مثالي تُعبّر عنها المعادلة الآتية:

${\mu }_B=\underset{B}{\overset{\ominus }{\mu }}+RT\ln x_B$

حيث $\underset{B}{\overset{\ominus }{\mu }}$ الكمون الكيميائي في الحالة المعيارية و$x_B$ الكسر المولي للمادة في المزيج.

وتُعمّم العبارة لتشمل السلوك الغير مثالي كالآتي:

${\mu }_B=\underset{B}{\overset{\ominus }{\mu }}+RT\ln a_B$

حيث$a_B$ الكسر المولي للمادة في المزيج:

$a_B=x_B{\gamma }_B$

حيث${\gamma }_B$ معامل الفاعلية.

وإذا اقتربت ${\gamma }_B$ إلى 1، يصبح سلوك المادة كأنّه مثالي، فيتحقق مثلًا قانون راؤول. وإذا كانت ${\gamma }_B>1$ أو ${\gamma }_B<1$ تُظهر المادة B انحرافًا موجبًا أو سلبيًا عن قانون راؤول (بالتوالي). ويدُلّ الانحراف الموجب على ازدياد تطايرية المادة B.

In many cases, as $x_B$ goes to zero, the activity coefficient of substance B approaches a constant; this relationship is Henry's Law for the solvent. These relationships are related to each other through the Gibbs-Duhem equation. Note that in general activity coefficients are dimensionless.

Modifying mole fractions or concentrations by activity coefficients gives the effective activities of the components, and hence allows expressions such as قانون راؤول and ثابت توازنs constants to be applied to both ideal and non-ideal mixtures.

Knowledge of activity coefficients is particularly important in the context of كيمياء كهربائية since the behaviour of كهرل solutions is often far from ideal, due to the effects of the ionic atmosphere. Additionally, they are particularly important in the context of soil chemistry due to the low volumes of solvent and, consequently, the high concentration of كهرل.

Application to chemical equilibrium

At equilibrium, the sum of the chemical potentials of the reactants is equal to the sum of the chemical potentials of the products. The طاقة غيبس الحرة change for the reactions, ${\mathrm{\Delta }}_{r}G$, is equal to the difference between these sums and therefore, at equilibrium, is equal to zero. Thus, for an equilibrium such as

$\alpha A+\beta B\rightleftharpoons \sigma S+\tau T$

${\mathrm{\Delta }}_{r}G=\sigma {\mu }_S+\tau {\mu }_T-(\alpha {\mu }_A+\beta {\mu }_B)=0$

Substitute in the expressions for the chemical potential of each reactant:

${\mathrm{\Delta }}_{r}G=\sigma \underset{S}{\overset{\ominus }{\mu }}-\sigma RT\ln a_S+\tau \underset{T}{\overset{\ominus }{\mu }}-\tau RT\ln a_T-(\alpha \underset{A}{\overset{\ominus }{\mu }}-\alpha RT\ln a_A+\beta \underset{B}{\overset{\ominus }{\mu }}-\beta RT\ln a_B)=0$

Upon rearrangement this expression becomes

${\mathrm{\Delta }}_{r}G=(\sigma \underset{S}{\overset{\ominus }{\mu }}+\tau \underset{T}{\overset{\ominus }{\mu }}-\alpha \underset{A}{\overset{\ominus }{\mu }}-\beta \underset{B}{\overset{\ominus }{\mu }})+RT\ln \frac{a_S^{\sigma }{a}_T^{\tau }}{a_A^{\alpha }{a}_B^{\beta }}=0$

The sum $(\sigma \underset{S}{\overset{\ominus }{\mu }}+\tau \underset{T}{\overset{\ominus }{\mu }}-\alpha \underset{A}{\overset{\ominus }{\mu }}-\beta \underset{B}{\overset{\ominus }{\mu }})$ is the standard free energy change for the reaction, ${\mathrm{\Delta }}_r{G}^{\ominus }$. Therefore

${\mathrm{\Delta }}_r{G}^{\ominus }=-RT\ln K$

K is the ثابت توازن. Note that activities and equilibrium constants are dimensionless numbers.

This derivation serves two purposes. It shows the relationship between standard free energy change and equilibrium constant. It also shows that an equilibrium constant is defined as a quotient of activities. In practical terms this is inconvenient. When each activity is replaced by the product of a concentration and an activity coefficient, the equilibrium constant is defined as

$K=\frac{[S{]}^{\sigma }[T{]}^{\tau }}{[A{]}^{\alpha }[B{]}^{\beta }}\times \frac{\underset{S}{\overset{\sigma }{\gamma }}\underset{T}{\overset{\tau }{\gamma }}}{\underset{A}{\overset{\alpha }{\gamma }}\underset{B}{\overset{\beta }{\gamma }}}$

where [S] denotes the تركيز of S, etc. In practice equilibrium constants are determined in a medium such that the quotient of activity coefficient is constant and can be ignored, leading to the usual expression

$K=\frac{[S{]}^{\sigma }[T{]}^{\tau }}{[A{]}^{\alpha }[B{]}^{\beta }}$

which applies under the conditions that the activity quotient has a particular (constant) value.

Measurement and prediction of activity coefficients

UNIQUAC تحليل الانحدار of Activity Coefficients (كلوروفورم/ميثانول Mixture)

Activity coefficients may be measured experimentally or calculated theoretically, using the Debye-Hückel equation or extensions such as Davies equation, Pitzer equations or TCPC model. Specific ion interaction theory (SIT) may also be used. Alternatively correlative methods such as UNIQUAC، NRTL، MOSCED or UNIFAC may be employed, provided fitted component-specific or model parameters are available.

A new alternative for activity coefficients prediction, which is less dependent on model parameters, is the COSMO-RS method. In this methods the required information comes from quantum mechanics calculations specific to each molecule (sigma profiles) combined with a statistical thermodynamics treatment of surface segments.

For uncharged species, the activity coefficient γ₀ mostly follows a "salting-out" model:

${\log }_{10}({\gamma }_0)=bI$

This simple model predicts activities of many species (dissolved undissociated gases such as CO₂, H₂S, NH₃, undissociated acids and bases) to high ionic strengths (up to 5 mol/kg). The value of the constant b for CO₂ is 0.11 at 10 °C and 0.20 at 330 °C.

For water (solvent), the activity a_w can be calculated using:

$\ln (a_w)=\frac{-\nu m}{55.51}$φ

where ν is the number of ions produced from the dissociation of one molecule of the dissolved salt, m is the molal concentration of the salt dissolved in water, φ is the osmotic coefficient of water, and the constant 55.51 represents the molal concentration of water. In the above equation, the activity of a solvent (here water) is represented as inversely proportional to the number of particles of salt versus that of the solvent.