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In an elementary chemical reaction, single molecules of two reactants A and B form a molecule of the product $ C: A + B \to C. $ The law of mass action states that the rate of reaction is proportional to the product of the concentrations of A and B:$ \frac {d[C]}{dt} = k [A][B] $(See Examples 3.7.4.) Thus, if the initial concentrations are $ [A] = a $ moles/L and $ [B] = b $ moles/L and we write $ x = [C], $ then we have $ \frac {dx}{dt} = k(a - x)(b - x) $(a) Assuming that $ a \neq b, $ find $ x $ as a function of $ t. $ Use the fact that the initial concentration of C is 0.(b) Find $ x(t) $ assuming that $ a = b. $ How does this expression for $ x(t) $ simplify if it is known that $ [C] = \frac {1}{2}a $ after 20 seconds?

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Calculus 2 / BC

Chapter 9

Differential Equations

Section 3

Separable Equations

Baylor University

Idaho State University

Boston College

Lectures

13:37

A differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders. An ordinary differential equation (ODE) is a differential equation containing one or more derivatives of a function and their rates of change with respect to the function itself; it can be used to model a wide variety of phenomena. Differential equations can be used to describe many phenomena in physics, including sound, heat, electrostatics, electrodynamics, fluid dynamics, elasticity, quantum mechanics, and general relativity.

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In an elementary chemical …

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If, in Example 4, one mole…

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In a general second-order …

problem number 40. Ah, the X over DT is equal to K A minus X Times B minus minus acts which is equal to okay X minus a times X minus minus B So the axe over a K X minus, a times X minus B is equal to DT if we take the integration for the posts the bullfights, so we will use the partial functions. The partial functions which said that one over X minus a x minus b is equal to a over X minus same times x minus B over X minus minus being plus be over X minus B times X minus A over X minus minus A So ah, this is equal to a X minus. B plus Be X minus minus A ah over X minus A x minus minus B to ah one uber X minus. A times X minus B is equal to a X minus. B plus B X minus a over X minus a times X minus minus B. Since the denominator are equal on both tides, the dominators must be equal to so one is equal to Ah a X minus minus B plus B x minus a This is which in one, so we substitute as X is equal to a in equation an equation one. So a is equal to one over a minus minus being. And if we substitute by saying that this is X is equal to eight when X is equal to be, ah B is equal to one over B minus. B minus eight. So they're poor. So therefore ah one over X minus eight times X minus minus B is equal to one over a minus. Bi one over X minus a minus one over X minus minus B So continuing from here we left and the integration is we said, um, previously immigration off the X over key X minus E x minus B, which is equal to integration off the tee so we can say that one over K A minus. Bi integration off one X minus A A minus one over X minus B The axe, which is equal to the in decoration off. Ah, integration off BT this integration is equal to one over K A minus bi Ah Lin ah Len, Off X minus a minus Len X minus minus B, which is equal to t plus at the Constance. So ah, so we can simply for these, as we can say one over K A minus bi ah times len X minus a over x minus b which is equal to t plus plus e. So given that the initial concentration ah is zero, which means that at X is equal to zero. T is equal to zero. So some institute the initial condition toe find and the constants So we can say that one over que a minus being times lend so X is equal to zero. So Len zero minus a over a zero minus b which is equal to zero plus a plus e. So from here we can say that K is equal to one over, um, capital were see the same meaning sea or K capital is equal to one over K A minus minus b times Len A over Overbey ah substituted the venue armed. Ah, see or K? So we can say that Ah, we can say that one over a key. A minus bi At times Lynn X minus a over x minus minus B is equal toe t plus one over que a minus bi times Len a over uber be to simplify these. Um, it's oppression. We can say that, Len X minus a over X minus B is equal to ah kee a minus, bi Ah, Times D plus Len a over Overbey. Ah, so ah, Len X minus a over x minus B ah minus Len. Or plus Len be over a is equal to okay, a minus. Miners being times time steam to raise the power. Ah, on both times to the basil Be so we can get So we raise a power toe Um, the power off e So we can be that x minus a over X minus a minus b which is equal to a over B e power key a minus b times feet. But she is equation. Also one Tosa abstract one from poor finds We can subtract one from both eyes. So x minus a over X minus B minus X minus B over X minus B, which is one is equal to a over B E K A minus B times T minus, minus one. So from here we can say that X minus B will be eliminated. H minus base. So is being minus a over X minus B, which is equal toe a over B times E a key off a minus Bi dynasty a minus one. I think there is a broken for both sides. And so we think that it's a broker for both sides. So we can get X minus. B over B minus. A is equal to one over a over B E K A minus B minus one. It's all about the Bible sides by ah B minus eight so x so x minus. B is equal to B minus A over a over B e. Okay, a minus. B minus minus one. Eso From here, we can say that axe is equal to B minus a over a over B E to the power off K A minus B minus one plus plus K claws being started. So from here, we can I multiplied and the pool sides by, um, this is being from here we can multiply both sides by this term a over B so we can with the blind both sides and by ah eight times e with the power of K A minus bi T minus B over eight times a poor Okay, a minus bi times d ah minus minus B so x will be equal to ah X will be equal Ah, to be squared minus a B plus a B e to the power off key A minus B times d minus b squared over a times a was the eve of the power off key A minus bi dynasty a minus minus beep. We here We just need to simply find expression to get, um, a suitable expression for X. Ah, So again, this can be simply find in tow a times b times e to the power off the A minus B times t a minus a B over eight times e to the power off K A minus bi dynasty Ah, minus minus B ah, we must the blind and Herminator and a denominator by e to the power. Ah, each of the power off key BT Ok, Bt so the final answer for that would be a be times e to the power old key 80 minus e with the power k B t over a times E to the power off K 80 minus b times e to the power off key B p so X is equal to a times B Times E to the power off K A T minus e to the power off K B T over eight times a to the power off K 80 minus b times e to the power off K B T.

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