Problem 40 Hard Difficulty

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?

Answer

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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.