chemical reaction equilibria

CHEMICAL REACTION EQUILIBRIA

Smith et al. Chapter 13

KUVENESHAN MOODLEY, [email protected]

KUVENESHAN MOODLEY, 2015

INTRODUCTION

o There are two factors that decide if a chemical reaction is significant

1. The chemical composition of the reactants and products

2. The rate at which the reaction occurs

o In this section we study the thermodynamic relationship between the temperature and pressure at which a reaction occurs with the resulting composition at equilibrium.

oThe equilibrium composition is a result of the maximum conversion being achieved for a particular reaction at a fixed T and P.

oThe speed (rate) at which this reaction is achieved is not studied here.


WHAT IS EQUILIBRIUM?

o A system at a constant T, P with phases in chemical equilibrium is said to be in thermodynamic equilibrium.

WHAT IS A PHASE?o A phase is a region of space that is distinct in boundary, and has uniform composition.

Can only occur if any reaction present is also in equilibrium


OUTLINE1. We develop mass balance equations that describe the composition profile as a reaction

proceeds, (13.1) (single and multiple reactions) using a parameter (Xi)/ (Epsilon)- the extent of reaction. This parameter can also be related to the total Gibbs energy of the system. (13.2)

2. We then use the Gibbs energy changes of the system to rigorously formulate the energy balances of the reaction, (13.3-13.5). The reaction temperature is factored in here.

3. In 13.6 we see the relationship between the mass and energy balance at equilibrium, and how we can use the mass balance to solve for the reaction temperature or vice versa.

4. We then go through calculation procedures for the above mentioned calculations for single reactions in either gas, liquid systems (homogeneous) or gas-liquid systems (heterogeneous), 13.7.

5. In Section 13.9 we look at procedures to solve systems involving multiple reactions. This also includes a procedure for the “global” minimization of the total Gibbs energy (Gt) of a system- a sure-fire way to solve ALL G-optimization problems.

6. Section 13.8 was covered in MEB and will not be covered here. You are encouraged to read through it on your own. 13.10 will not be covered.


13.1 THE REACTION COORDINATEoWe provide here the rigorous thermodynamic development of the method. You will soon realise you have been doing this all along.

(13.1)

o For a reaction, the change in the number of moles of a component A1 is directly proportional to the change in the number of moles of A3. As A1 depletes, A3 increases. The proportionality constant is

oEqually the change in the number of moles of a component A1 is directly proportional to the change in the number of moles of A2. As A1 depletes, A2 depletes. The proportionality constant is


13.1 THE REACTION COORDINATE So the depletion of A1 :

And the depletion of A2:

And the growth of A3:

Now remember that is positive for products and negative for reactants

So is a and is a

As we said earlier, each proportional change is equal, therefore:


13.1 THE REACTION COORDINATE

Now if we want to rather examine a differential change in the number of moles we can say

(13.2) or for every component i in the reaction (13.3)

Now if we integrate from the start of the reaction (where no reaction has occurred, =0 ) to some arbitrary point where :

We get or (13.4)


EXERCISE 1: 3 MINUTES

o If the mole fraction, is given by definition as:

Determine a general expression for , in terms of , , and


EASILY:


13.1 THE REACTION COORDINATE Now for multiple reactions we must consider an additional step:

reaction 1

reaction 2

Then the change in the number of moles of component i is due to both reactions:

+

So = +

Or more generally , where i, refers to the component number and j refers to the number of reaction equations.


HOMEWORK 1

o If the mole fraction, is given by definition as:

Determine a general expression for , in terms of , , and


13.2 APPLICATION OF EQUILIBRIUM CRITERIA TO CHEMICAL REACTIONSo Like all real closed systems, a spontaneously reacting system at constant T and P will proceed in such a way to attain the lowest total Gibbs energy state. This occurs at a minimum H and a maximum S. This is a consequence of the 3 laws of thermodynamics.

o At equilibrium , P, yi, is constant and therefore, Gt must be a minimum.

oThis occurs at the equilibrium extent or for “equilibrium”.

oIf Gt is a minimum, or “turning point”, then any differential changes in the total Gibbs energy must be zero.

WHAT DOES THIS MEAN?oWe are therefore presented with two options for solving the point, at which minimum Gt occurs at constant T and P.

oWe can either 1. minimize Gt directly 2. or we can set

oOption 1 is the more rigorous calculation, and guarantees a solution (if it exists)

oOption 2 is a “shortcut” method, that employs the use of equilibrium constants. It is sometimes called the K method. This method can almost always be used for single reactions but sometimes fails in multi-reaction systems.



13.3 THE STANDARD GIBBS-ENERGY CHANGE AND THE EQUILIBRIUM CONSTANT 1. Start with the fundamental property relation (11.2).

2. If we assume that chemical equilibrium is only due to changes from reacting species then we can replacewith some relation to

3. T, P are fixed so and we get (13.8)


13.3 THE STANDARD GIBBS-ENERGY CHANGE AND THE EQUILIBRIUM CONSTANT 4. Now work on eliminating the chemical potential given by (11.42) (which is immeasurable).

5. Since we know (11.30) and we can define a standard state at a particular temperature and pressure then we get:

6. Now subtract this from (11.42) and we get (13.9)

7. Eliminating in equation (13.8)


13.3 THE STANDARD GIBBS-ENERGY CHANGE AND THE EQUILIBRIUM CONSTANT 8. Now simplify

9. We define K to represent:

Where:

10. For a reacting system by definition:

11. Hence


13.3 THE STANDARD GIBBS-ENERGY CHANGE AND THE EQUILIBRIUM CONSTANT 12. All property relations apply to standard property changes of reaction. Hence the standard enthalpy change of reaction can be related to the standard Gibbs energy change of reaction:


13.4 EFFECT OF TEMPERATURE ON THE EQUILIBRIUM CONSTANTo Despite being called the equilibrium constant, K, is in fact strongly influenced by temperature.

oWe can calculate this temperature dependence by finding the derivative of lnK with respect to T.

oHence (13.14)

oA check confirms that for a reaction that is exothermic (product has less enthalpy than reactant), and that an increase in T will cause a reduction in K.

oAnd that if then an increase in T will cause an increase in K.

13.4 EFFECT OF TEMPERATURE ON THE EQUILIBRIUM CONSTANTo Equation 13.14 can be integrated easily if the standard enthalpy change of reaction is assumed to not vary with T.

o If it IS dependent on T however, we must use a more rigorous method to integrate the preceding equation for K.

oWe start with the G axiom for standard changes of reaction:

oThen apply our rule for property change of reaction

oSimplifying KUVENESHAN MOODLEY, 2015

∆𝐺°=∆𝐻°−𝑇 ∆𝑆°


13.4 EFFECT OF TEMPERATURE ON THE EQUILIBRIUM CONSTANToWork on , which was covered previously (4.18)

o And if we use the property change of reaction tool we can get from (13.17)

oCombine and simplify (13.18)


13.4 EFFECT OF TEMPERATURE ON THE EQUILIBRIUM CONSTANT

Now if we multiply by -1and take logarithms we get

ln(K0) ln(K1) ln(K2)ln(K)

o So

o And


13.4 EFFECT OF TEMPERATURE ON THE EQUILIBRIUM CONSTANTo And K2 which accounts for the two integrals, is (13.24):

Where

o From the log rules we can therefore say (13.20):


HOMEWORK 2

1. From the Cp integral equations provided in your text, show by rigorous calculation that equation (13.24) is valid.

2. If determine the equation for K2

3. Compare with friends

4. Perform example 13.5 from the textbook. Ensure you follow all the calculations and obtain the solutions given. DO NOT USE EXCEL.


13.6 RELATION OF EQUILIBRIUM CONSTANTS TO COMPOSITIONoGas phase reactions (homogenous)

oLiquid phase reactions (homogenous)

oMixture of gas and liquid (heterogeneous)


ELIMINATING FUGACITY IN SOLUTION ()• We defined the relationship between

the fugacity in solution, standard fugacity and K in Section 13.3 (13.10)

• If we choose a reference state as the ideal gas state, then the component fugacity if equal to the pressure at that reference temperature. We get this reference pressure from the same table we acquire standard property changes of reaction. (13.25)

• Now what we do is use the relationships between fugacity in solution (), fugacity coefficient in solution (), and composition (yi or xi ) to eliminate fugacity in solution


SUMMARY OF REPLACEMENTSCondition Gas Phase Liquid phase

Partial residual Gibbs energy

Where by definition:

Partial excess Gibbs energy


Real

Ideal solution

Ideal Gas N/A

𝛾𝑖𝑖𝑑=1

𝜙𝑖𝑖 𝑔=1


WHAT ARE THE EQUILIBRIUM EQUATIONS FOR EACH TYPE OF REACTION?Condition Gas Phase Liquid phase

Partial residual Gibbs energy


Partial excess Gibbs energy


Real

Ideal solution

Ideal Gas N/A

𝛾𝑖𝑖𝑑=1

𝜙𝑖𝑖 𝑔=1

Complete this table for Homework 3


ELIMINATING FOR LIQUIDS

(6.10)

Integrate 6.10 at constant T from stand pressure to system pressure

chemical reaction equilibria

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