Arrhenius Equation

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I tried this question from the MCATQuestionaday website:

Q: The Arrhenius equation, shown below, is a description of the relationship between the rate constant, k, the activation energy of the reaction, and the temperature in Kelvin.

k = Ae^-Ea/RT

Here, A and R are constants. If the constant R were half of its actual value, what would happen to the rate constant k? Note: assume the value of e is 2.71.

A It would increase by less than double.
B It would increase by more than double.
C It would decrease to less than half of its current value.
D It would decrease to more than half of its current value.

Why? Since we are concerned primarily with what is happening to R in the reaction, we can ignore the remaining variables and notice that if R doubles, then the value of -EA/RT is being halved. Thus the question comes down to what happens when we have e-1/2, or, expressed another way, 1/√e. Given the value of e as 2.71, we can estimate its square root to be roughly 1.6 (note that 1.52 = 2.25 and 1.72 = 2.89. The inverse of 1.6 is roughly 0.6 meaning the value of the rate constant would decrease but would remain greater than half its current value, making (D) the correct answer.

I am confused since the question is saying what if R becomes half. I thought that if R is halved, then the exponent on e is doubled (because R is in the denominator of the exponent) and k would increase by more than double.

Can someone explain this please?

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[QrtrLifeCrisis]

So, I narrowed it down to C or D, but I can't seem to figure out why it would be more or less than half.

R is in the denominator, so when you halve it, you're effectively doubling the whole exponent expression. e^-Ea/(.5)RT = e^-2Ea/RT = (e^-(Ea/RT))^2. Obviously, this number is less than the original number, but this is as far as I could get. I don't think you know whether it would decrease by more or less than half without knowing the actual values involved. I could be wrong.

Edit:
I just took a look at the actual website and there is something fishy about the answer. They speak as if R was doubled, but the question asks about R being halved. Also, the way they pulled e^-1/2 out is incorrect because that doesn't follow the rule multiplying exponents.

Any other thoughts?

 

[AureatEagle]

I tried this question from the MCATQuestionaday website:

Q: The Arrhenius equation, shown below, is a description of the relationship between the rate constant, k, the activation energy of the reaction, and the temperature in Kelvin.

k = Ae^-Ea/RT

Here, A and R are constants. If the constant R were half of its actual value, what would happen to the rate constant k? Note: assume the value of e is 2.71.

A It would increase by less than double.
B It would increase by more than double.
C It would decrease to less than half of its current value.
D It would decrease to more than half of its current value.

Why? Since we are concerned primarily with what is happening to R in the reaction, we can ignore the remaining variables and notice that if R doubles, then the value of -EA/RT is being halved. Thus the question comes down to what happens when we have e-1/2, or, expressed another way, 1/√e. Given the value of e as 2.71, we can estimate its square root to be roughly 1.6 (note that 1.52 = 2.25 and 1.72 = 2.89. The inverse of 1.6 is roughly 0.6 meaning the value of the rate constant would decrease but would remain greater than half its current value, making (D) the correct answer.

I am confused since the question is saying what if R becomes half. I thought that if R is halved, then the exponent on e is doubled (because R is in the denominator of the exponent) and k would increase by more than double.

Can someone explain this please?

edit: messed up the math

 

[AureatEagle]

So, I narrowed it down to C or D, but I can't seem to figure out why it would be more or less than half.

R is in the denominator, so when you halve it, you're effectively doubling the whole exponent expression. e^-Ea/(.5)RT = e^-2Ea/RT = (e^-(Ea/RT))^2. Obviously, this number is less than the original number, but this is as far as I could get. I don't think you know whether it would decrease by more or less than half without knowing the actual values involved. I could be wrong.

Edit:
I just took a look at the actual website and there is something fishy about the answer. They speak as if R was doubled, but the question asks about R being halved. Also, the way they pulled e^-1/2 out is incorrect because that doesn't follow the rule multiplying exponents.

Any other thoughts?

I agree-- they messed up. They meant to ask what would happen if you doubled R, and even then they did the math incorrectly. I don't believe you can tell.