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GMAT Prep - THe rate of a reaction is directly proportional.

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GMAT Prep - THe rate of a reaction is directly proportional.

by barrelbowl » Sat Dec 17, 2011 12:39 pm
The rate of a reaction is directly proportional to the square of the concentration of A and inversely proportional to the concentration of B. If B increases by 100% which of the following is closest to the % change in the concentration of A required to keep the rate unchanged?

A. 100% decrease
B. 50% decrease
C. 40% decrease
D. 40% increase
e. 50% increase

OA: D

I have a question regarding proportionality. I searched the former for the solution and was a little confused.

Are we allowed to set up the equations, with "R" as the rate of reaction and "k" and "z" as the proportionality constants.

R = k*x^2

and

R = (1/z)*B

And assume that because R is the common term, that Z = K? So we can combine the equations into R=k(x^2)/B?
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by barrelbowl » Sat Dec 17, 2011 1:23 pm
Ahh sorry screwed up the second equation.

R = Z/B not R = (1/z)*B
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by neelgandham » Sat Dec 17, 2011 1:25 pm
The rate of a reaction is directly proportional to the square of the concentration of A and inversely proportional to the concentration of B. Implies
R = K * (A*A)/B, Where R is the rate of the reaction, K is a Constant, A is concentration of A and B is concentration of B.
If the value of B Increases by 100%,
The new rate of reaction, R1 = K * (A*A)/2B
But it is also given that the rate of reaction is left unchanged, i.e if the denominator of the fraction K * (A*A)/B increases 2 folds then the numerator(A^2) should also increase 2 times the original value.Options A, B, C are eliminated(They say decrease).

A^2/B should remain constant

A^2/B = A1^2/2B, Where A1 is the new concentration of A
A1^2 = 2A^2

From option e) A1 = 1.5A, Implies A1^2 = 2.25 A^2. May be the answer? Let us check option D
From option d) A1 = 1.4A, Implies A1^2 = 1.96 A^2. Yes this is the answer, becuase 1.96A^2 is closer to 2 A^2 than is 2.25A^2

I hope this answers all you questions regarding setting up of equations ! Let me know if you still need help with this.
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by GmatMathPro » Sat Dec 17, 2011 3:14 pm
barrelbowl wrote: Are we allowed to set up the equations, with "R" as the rate of reaction and "k" and "z" as the proportionality constants.

R = k*x^2

and

R = (1/z)*B

And assume that because R is the common term, that Z = K? So we can combine the equations into R=k(x^2)/B?
Think about it this way:

When you write something like R=k*A^2, it almost looks the rate of reaction depends solely on A. But of course in the real world, it depends on lots of different variables (like temperature, pressure, the presence of catalysts, the concentration of other reactants, etc.). So how can we legitimately express the rate of reaction so simply? It's because we're assuming that all of these other variables that affect it are being held constant. So that constant, k, is capturing all of that different information and combining it into one number. For example, the real formula for the rate of reaction might be something really complex like R=z*C*T*P*A^2/B. But if we're holding everything except for A constant, then we can go ahead and plug all the known values into the other variables, which will give us some number, and then that constant is accounting for all these other factors, so we can say z*C*T*P/B=k, and then we can rewrite the equation as R=k*A^2. Or, if everything except for B is being held constant, we could say z*C*T*P*A^2=k, and then we can say R=k/B.

So don't think of it so much as two separate equations that you have to find some justification for combining into one. It's more natural to think of it as one big master equation that takes into account ALL of the factors. But that huge equation can be simplified down to equations in terms of one variable by holding everything else constant.

In the future, if you see a problem like this where some quantity depends on several things, and you decide you want to model it with equations, you should just start out by writing one equation for it with one constant of proportionality that includes all of the variables relevant to the problem.
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by GMATGuruNY » Sat Dec 17, 2011 3:30 pm
barrelbowl wrote:The rate of a reaction is directly proportional to the square of the concentration of A and inversely proportional to the concentration of B. If B increases by 100% which of the following is closest to the % change in the concentration of A required to keep the rate unchanged?

A. 100% decrease
B. 50% decrease
C. 40% decrease
D. 40% increase
e. 50% increase

OA: D

I have a question regarding proportionality. I searched the former for the solution and was a little confused.

Are we allowed to set up the equations, with "R" as the rate of reaction and "k" and "z" as the proportionality constants.

R = k*x^2

and

R = (1/z)*B

And assume that because R is the common term, that Z = K? So we can combine the equations into R=k(x^2)/B?
We can solve this problem by plugging in values. Check here:

https://www.beatthegmat.com/the-rate-of- ... 60321.html
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by barrelbowl » Sun Dec 18, 2011 8:08 am
Thanks guys. GMATMathPro - that explanation about how multiple variables affect the proportionality constant was especially helpful.
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