Extent of reaction

In physical chemistry and chemical engineering, extent of reaction is a quantity that measures the extent to which the reaction has proceeded. Often, it refers specifically to the value of the extent of reaction when equilibrium has been reached. It is usually denoted by the Greek letter ξ. The extent of reaction is usually defined so that it has units of amount (moles). It was introduced by the Belgian scientist Théophile de Donder.

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In this screencast we are going to go through an example, were we solve material balances involving multiple reactions using the extent of reaction method. Now what is the extent of reaction method? We write it as zeta, t has units of moles, and we can use it with reactions to determine the extent at which the reaction proceeds. Now lets look at our general definition for our material balances. In which we have no accumulation. So we say that in plus generation minus consumption is what comes out. Now if we write this as the moles of a reactor are equal to the moles in. When we group generation and consumption gives us an idea on how many reactions have occurred and we can write this in stoichiomtric coefficient of each species times the extent of reaction. So lets take a quick look at a simple reaction. A going to 2B. Were we can write this as the moles of A are going to be equal to the moles of the A that we start with plus or minus depending what side of the equation that we are on. Since we are consuming A we are going to subtract out 1 mole for every reaction that occurs on a molar basis. For B,same initial set up, but now we are generating 2 B molecules for every A molecule that has reacted. So now we have used the extent of reaction right, 2 material balances. For both A and B. Now lets take a look at the problem at hand. So the following problem says. We have 100 moles of ethane are sent to a reactor and react to form ethylene and hydrogen gas. However a side reaction occurs, which ethylene reacts with ethane to form propylene and methane. Now we are told the fractional conversion rate of methane is 0.7, and the selectivity of propylene to ethylene is 5, and the goal is to find the composition of the product gas. So with any material and energy balance problem we should start by drawing a figure. So here I have my reactor with both reactions in it, and we know that some amount of ethane comes in 100 moles and leaves as a product stream, and we are asked to determine the composition of that product stream. So we need to finish labeling or system by putting in our unknowns. We should also write in our fractional conversion. Since it is given to us, and our selectivity of ethylene to propylene, and that is given as 5. So we have all our that we have been given in the statement, and we have all the information that we need to find labeled. The next logical step is do to a degree of freedom analysis to make sure we are not missing anything. So for a degree of freedom analysis we start by with our unknowns, which we have 5 above. We add the amount of independent reactions. We have 2 reactions. We subtract out the number of molecular species, and you can see from our 2 reactions we have 5 species and we subtract out any other pieces of information that can help us solve our problem, and we have selectivity and conversion. When we add the first 2 up and subtract the bottom 2 we have a degree of freedom analysis that gives us 0, which means that it is solvable as is. So we have written all our known and unknowns so now we can move one. So lets start by using the fractional conversion, which says that is value 0.7 is going to be the amount that has converted. From what has come in to what has come out into our reactions. So we can write this by saying that what goes in to our reactor minus what leaves the reactor divided by what went in to the reactor is equal to the 0.7, and our selectivity tells us the ratio of the amount of moles of C2H4 to the undesired product propylene. So those are our 2 pieces of information that we are going to use with our extent of reaction equations. You can use the extent of reaction as you recall, were we write the amount of moles leaving the reactor for our first product. That is going to be equal to the amount the originally went in. We designate that with a 0 at the bottom and we are going to subtract out the extent of reaction for reaction 1. Since for every reaction that occurs we lose 1 mole and since it reacts in reaction 2. We are going to subtract the extent of reaction 2 and that would be our first equation, and we repeat this for our other products. Write it as what came in and since we are producing it in the first reaction. We write it as a plus and we are using in in the second reaction or consuming it. So we subtract out the extent of reaction 2. Now I am going to continue this for the other 3 species and we have our 5 balances for our 5 species. So you can count the number of unknowns we have. So we have an known here 2,3,4,5, and our extent of reaction for the 1 and 2 are unknown, but we know that is entering our system, and that is given in the problem statement, and we know that is the only thing entering. So we can get rid of this term. So we are down to 7 unknowns. 5 reactions equations, and 2 pieces of information we have above it. So we can move on to solve. Now take a look at our first equation and substitute what we know here, which we have 100 moles entering our processes, and we know that was the only thing entering. So this should go away. So we can start solving based on the information we have. You can see that this in the only we have to solve for using our fractional conversion. So solving for the moles leaving we get 30, and plug that in here. We can also rearrange the first equation so it looks like the following. We can use our selectivity and plug in our equations, which would be equation 2 and equation 4. into our selectivity equation. Doing so looks like the following. Now we can use that equation to have a relationship between the extent of reaction 1 and the extent of reaction 2, and we can see that the extent of reaction 1 is 6 times that of reaction 2. We are going to plug this information into equation 1. Solve for the extent of reaction 2, equaling 10 moles, and using what we had before extent of reaction 1 is equal to 60 moles. So we have solved for both of our extent of reactions, and we can plug that information into our 5 equations as you see here. Now we can solve for all of our unknowns, and hopefully you get the same values I got. Since the problem was asking what the composition of the product would be at this point we would need to determine the molar fractions of each component. We can add up all of these, and we get a total amount of moles of 160. Now to solve for each of the molar fractions we will designate as y. Just the amount of moles of the species we are interested in divided by the total amount. So here I have written out the molar fractions of each of the components, and a good thing to do is check that they add up to 1. Hopefully you can see through this video how to use the extent of reaction to calculate for unknowns in a material balance involving multiple reactions, and you can see the more reactions we have the more complicated the problem gets as you introduce a new variable for each reaction.

Definition

Consider the reaction

A ⇌ 2 B + 3 C

Suppose an infinitesimal amount $dn_{i}$ of the reactant A changes into B and C. This requires that all three mole numbers change according to the stoichiometry of the reaction, but they will not change by the same amounts. However, the extent of reaction $\xi$ can be used to describe the changes on a common footing as needed. The change of the number of moles of A can be represented by the equation $dn_{A}=-d\xi$ , the change of B is $dn_{B}=+2d\xi$ , and the change of C is $dn_{C}=+3d\xi$ .^[1]

The change in the extent of reaction is then defined as^[2]^[3]

d\xi ={\frac {dn_{i}}{\nu _{i}}}

where $n_{i}$ denotes the number of moles of the $i^{th}$ reactant or product and $\nu _{i}$ is the stoichiometric number ^[4] of the $i^{th}$ reactant or product. Although less common, we see from this expression that since the stoichiometric number can either be considered to be dimensionless or to have units of moles, conversely the extent of reaction can either be considered to have units of moles or to be a unitless mole fraction.^[5] ^[6]

The extent of reaction represents the amount of progress made towards equilibrium in a chemical reaction. Considering finite changes instead of infinitesimal changes, one can write the equation for the extent of a reaction as

\Delta \xi ={\frac {\Delta n_{i}}{\nu _{i}}}

The extent of a reaction is generally defined as zero at the beginning of the reaction. Thus the change of $\xi$ is the extent itself. Assuming that the system has come to equilibrium,

\xi _{equi}={\frac {n_{equi,i}-n_{initial,i}}{\nu _{i}}}

Although in the example above the extent of reaction was positive since the system shifted in the forward direction, this usage implies that in general the extent of reaction can be positive or negative, depending on the direction that the system shifts from its initial composition.^[7]

Relations

The relation between the change in Gibbs reaction energy and Gibbs energy can be defined as the slope of the Gibbs energy plotted against the extent of reaction at constant pressure and temperature.^[1]

\Delta _{r}G=\left({\frac {\partial G}{\partial \xi }}\right)_{p,T}

This formula leads to the Nernst equation when applied to the oxidation-reduction reaction which generates the voltage of a voltaic cell. Analogously, the relation between the change in reaction enthalpy and enthalpy can be defined. For example,^[8]

\Delta _{r}H=\left({\frac {\partial H}{\partial \xi }}\right)_{p,T}

Example

The extent of reaction is a useful quantity in computations with equilibrium reactions. Consider the reaction

2 A ⇌ B + 3 C

where the initial amounts are $n_{A}=2\ {\text{mol}},n_{B}=1\ {\text{mol}},n_{C}=0\ {\text{mol}}$ , and the equilibrium amount of A is 0.5 mol. We can calculate the extent of reaction in equilibrium from its definition

\xi _{equi}={\frac {\Delta n_{A}}{\nu _{A}}}={\frac {0.5\ {\text{mol}}-2\ {\text{mol}}}{-2}}=0.75\ {\text{mol}}

In the above, we note that the stoichiometric number of a reactant is negative. Now when we know the extent, we can rearrange the equation and calculate the equilibrium amounts of B and C.

n_{equi,i}=\xi _{equi}\nu _{i}+n_{initial,i}

n_{B}=0.75\ {\text{mol}}\times 1+1\ {\text{mol}}=1.75\ {\text{mol}}

n_{C}=0.75\ {\text{mol}}\times 3+0\ {\text{mol}}=2.25\ {\text{mol}}

References

^ ^a ^b Atkins, Peter; de Paula, Julio (2006). Physical chemistry (8 ed.). p. 201. ISBN 978-0-7167-8759-4.
^ Lisý, Ján Mikuláš; Valko, Ladislav (1979). Príklady a úlohy z fyzikálnej chémie. p. 593.
^ Ulický, Ladislav (1983). Chemický náučný slovník. p. 313.
^ IUPAC, Compendium of Chemical Terminology, 2nd ed. (the "Gold Book") (1997). Online corrected version: (2006–) "stoichiometric number, ν". doi:10.1351/goldbook.S06025
^ Canagaratna, Sebastian C. (January 1, 2000). "The Use of Extent of Reaction in Introductory Courses". J. Chem. Educ. 77 (1): 52. doi:10.1021/ed077p52. Retrieved 3 May 2021.
^ Hanyak, Jr., Michael E. "Extent of Reaction or Events of Reaction?" (PDF). Department of Chemical Engineering, Bucknell University. 2014. Retrieved 3 May 2021.
^ Vandezande, Jonathon E.; Vander Griend, Douglas A.; DeKock, Roger L. (August 23, 2013). "Reaction Extrema: Extent of Reaction in General Chemistry". Journal of Chemical Education. 90 (9): 1177–1179. doi:10.1021/ed400069d. Retrieved 10 July 2021.
^ Lisý, Ján Mikuláš; Valko, Ladislav (1979). Príklady a úlohy z fyzikálnej chémie. p. 593.

This page was last edited on 14 March 2023, at 20:27

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