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Some reactions are described as parallel in that the reactant simultaneously forms different products with different rate constants. An example is and $$\begin{aligned}&A \stackrel{k_{1}}{\longrightarrow} B\\&A \stackrel{k_{2}}{\longrightarrow} C\end{aligned}$$ The activation energies are $45.3 \mathrm{kJ} / \mathrm{mol}$ for $k_{1}$ and $69.8 \mathrm{kJ} / \mathrm{mol}$ for $k_{2} .$ If the rate constants are equal at 320 $K,$ at what temperature will $k_{1} / k_{2}=2.00 ?$

$$298 \mathrm{K}$$

Chemistry 102

Chapter 6

Chemical Kinetics

Kinetics

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Drexel University

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Okay, So this question they tell us that the activation energy of K one is 45.3 killed. Jos have converted that into jewels here. And they tell us that the activation energy of K two is 69.8 till jewels have also converted. Thanks rituals. And they tell us that the K one is equal to K two at 320 k Right? So the first thing that we can do is we need to know this equation. There really isn equation. Okay, It's good. A e to the negative e a over our tea, right? E a is the activation energy activation and she are is the gas constant 8.3154 Joe's Pirlo Kevin and T is the temperature Didn't Kevin and a is the concentration, right? So the first thing that we can do is we can find the ratio of the A one and a two at 320 camp, right? Since we know that K one is equal to K to this means that a one p to negative activation energy over r t. It's equal to a two e negative activation energy over our teeth, right? Because K one, let's say this is K one and we have K two is the exact same equation and they're equal. We can set him equal to each other. So this is me setting them equal to each other. Now who can dio, as we can plug in our values and solve for the racial of a one over a two. Right. So what we can do is we do a one e to the negative. 45300 Jules Pramono, I'm not gonna write the units. So? So it won't be too messy. Divided by 8.3 warned 54 times it by 320 K You're under 20 K And that is equal to a to E to the negative. Six 9800 That is the second activation energy over G gas. Constant times that by 320. All right. And we get a one over a two as 1.0 two times 10 to the negative for as the ratio. No, we should we can use this ratio that we have for Sorrenti. Question. Right. So they tell us that, uh, we need to find the temperature at which k one over K two. So the ratio is equal to and since we know the ratio at 320 k is 1.2 time sensitive they get for we can use that value. Right. So we have a one e they got a e a one over are tee over a to e minus e a two over rt is able to to eso we plug in a 1/80 which is 1.2 times 10 to the negative four this is equal to And then we have e activation energy of one over r t over E to the activation energy. Negative activation energy of two over rt goes to right. So now we can do is it can isolate t from, uh, this equation. So what I'm gonna do now is I want to multiply. Now we're going to divide both sides. I won't 0.2 times 10 to the negative for about one point ill so two times 10 to the negative four and we get e to the negative e a one over r t plus e a to over our key is equal to 19959 Now what we can dio is we take the Ellen on both sides. So we get rid of Oh, So the reason why I got this is because when we said track when we divide exponents, we do the exponents on the top. So the top exponents minus t bottom exponents Minister Bonham X phone a insisted bottom exponents is negative. S 02 negatives means that two minus negative bottom exponents means is top exponents plus the bottom exponents. That's why the other is a positive sign here. But now we take the Ellen of both sides. So we get rid of E because natural law cancels out e So we get negative. He a one over tea plus the activation energy of two over r t physical to L and 19959 Now, since we're adding these two and they have the same denominator, you know that we can write that as e A to minus e a one right, because if I distribute the negative into the parentheses and I really, really organized it, weaken. But as that over r t is equal to Ln 19959 I said, What I can do now is like a multiply both sides by t e a to e for our secret to health. And 19959 times it by t. Now we divide by Ln 1995 on both sides, and we get t by itself. So now you plug Andy values to get t is equal to activation. Energy of two is acquitted. 69800 Jewels promote minus 45300 Jewels promo over 8.31 for five tools. Promote coven times it by the natural laws of 19959 Should get the answer is 298 Kevin. So at 298 Kelvin do you ratio ratio of K one over K two is equal to two.

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