Extent of Reaction for Material Balances

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.