Chemical Forums
Chemistry Forums for Students => High School Chemistry Forum => Topic started by: Big-Daddy on August 25, 2013, 12:54:10 PM
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t (/ms); [HCl] (/mol·dm-3)
2; 0.57
4; 1.05
8; 1.74
12; 2.19
16; 2.46
∞; 3.00
Given this experimental data, and the function for k1 in terms of [HCl] and [HCl]t=∞, how can I calculate a best possible value for k1? Obviously just choosing any of the numbers and plugging them in gives me one value, but that value doesn't correspond exactly to producing the other numbers.
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Probably some sort of linear regression or function fitting algorithm. Origin will do this without breaking a sweat. Or you can always do it by hand.
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How accurate would it be just to get all the k_1 values (at each t) and then take the average?
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Does the problem specify what the rate law is?
Averaging would probably be better than nothing, but I suspect that it would be inferior to linear or nonlinear regression. For one thing, not all of the data points are likely to be equally well determined, and using statistical weights can counteract this problem (although the issue of weighting the data is probably beyond what you need to worry about for right now). Here are two reviews on the subject of nonlinear regression:
http://www.cell.com/trends/biochemical-sciences/abstract/0968-0004(90)90295-M
http://www.fasebj.org/content/1/5/365.abstract
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Why don't you at least write down the "function for k1 in terms of [HCl] and [HCl]t=∞" part for us?
Please. Thanks.
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Because, as always with me, this is a more general problem than that. ;D I'm not worried about the actual calculation of k1, just the fact that I'd end up with a different one for each data reading.
Anyway for your interest here goes:
[tex]c_{HCl} = c_{HCl,t = \inf} \cdot (1 - e^{-k \cdot t \cdot c_{CH4}})[/tex]
[CH4] was given. And now solving for k1 is almost painfully easy, just rearrange. But getting a good average is the problem.
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Does the problem specify what the rate law is?
Averaging would probably be better than nothing, but I suspect that it would be inferior to linear or nonlinear regression. For one thing, not all of the data points are likely to be equally well determined, and using statistical weights can counteract this problem (although the issue of weighting the data is probably beyond what you need to worry about for right now). Here are two reviews on the subject of nonlinear regression:
http://www.cell.com/trends/biochemical-sciences/abstract/0968-0004(90)90295-M
http://www.fasebj.org/content/1/5/365.abstract
Thanks for the advice. I'm guessing linear regression would have been ok for this problem and that's what I'll have a look at. (I studied it a few months ago but forgot after my exams, I'm not that keen on statistics :D )
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Do you know how to write your equation in a linear form?
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Because, as always with me, this is a more general problem than that. ;D I'm not worried about the actual calculation of k1, just the fact that I'd end up with a different one for each data reading.
Anyway for your interest here goes:
[tex]c_{HCl} = c_{HCl,t = \inf} \cdot (1 - e^{-k \cdot t \cdot c_{CH4}})[/tex]
[CH4] was given. And now solving for k1 is almost painfully easy, just rearrange. But getting a good average is the problem.
If I were setting the problem I'd insert an outlier point to make a brute force average blow up and totally mess your answer.
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If I were setting the problem I'd insert an outlier point to make a brute force average blow up and totally mess your answer.
I don't think this is actually in the problem, but there'd be no point anyway - my calculator cannot handle the entire expression being put in 6 times, so I'd be calculating k1 values one-by-one and taking the average. Obviously I'd eliminate clear anomalies. But it's a matter of principle really. Given the rest of the question was so easy (rearrangement) and it's worth more marks than the part that asks you to actually derive this equation from the reaction mechanism, I'm guessing they want regression of some form.
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Do you know how to write your equation in a linear form?
What do you mean? I might give the rearrangement a go but what are you suggesting I rearrange for exactly?
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@BD.
Just for your own benefit: this is why just calculating k's from single points and averaging is not a good way to do things.
Using your data I calculate k*[CH4] of 0.108 using your "averaging method" and 1.0979 using a basic linear regression with partial least squares. Not too bad, right?
Ok, well change your first point (t = 2) from 0.57 to 0.17. Using the averaging method now I get a value of 0.09328, a drop of about 9% or so. Using the PLS regression, I get a value for k*[CH4] of 1.0917, a deviation of <1%.
Even a simple regression can tolerate an anomalous outlier point, whereas just averaging the values cannot, because it weights each point evenly.
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Ok, thanks for the outline.
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Do you know how to write your equation in a linear form?
What do you mean? I might give the rearrangement a go but what are you suggesting I rearrange for exactly?
For linearity. y=kx+c
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And here, k1 is y, t is x?
I can rearrange it for k1 = k/t where k is a constant.
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And here, k1 is y, t is x?
I can rearrange it for k1 = k/t where k is a constant.
y is the dependent variable. x is the independent. c and k are clubbed together constants.
If you even suspected k1 is y you need to revisit fundamentals of curve fitting or regression I think.
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Big-Daddy,
In my experience students have a hard time getting the notion that in regression, the parameters (c and k) are what the experimenter is usually most interested in, as opposed to the variables (x and y). Perhaps you could look at your equation and identify the parameters and the variables.
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Big-Daddy,
In my experience students have a hard time getting the notion that in regression, the parameters (c and k) are what the experimenter is usually most interested in, as opposed to the variables (x and y). Perhaps you could look at your equation and identify the parameters and the variables.
The confusing semantics of "parameters, variables and constants" is another source of misunderstanding.
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Can you elaborate on this? Under normal circumstances the independent variable should have little or no error, but the dependent variable may have error with certain caveats. Often the experimenter can control the independent variable and measure the dependent variable. That leaves us with what to call c and k in this example. Perhaps I should have not used the word parameters (so that things would be less muddled).
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Can you elaborate on this? Under normal circumstances the independent variable should have little or no error, but the dependent variable may have error with certain caveats. Often the experimenter can control the independent variable and measure the dependent variable. That leaves us with what to call c and k in this example. Perhaps I should have not used the word parameters (so that things would be less muddled).
To me independent and dependent are clear ways of writing it. But I've seen "parameter" being used in so many ways that I'm always confused.
The problem a novice faces with a "constant" is he assumes it is a "constant" so kind of counter-intuitive that the regression is trying to determine his "constant". i.e. If you ever have to manually do a least-squares regression the constants are variable and the variables are multiple constants. ;D
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It's a constant in that it's not being varied as part of the experimental design. You have the independent variable, which is the experimental variable; you have the dependent variable, which is the quantity you measure as a function of the independent variable; and you have a series of parameters, constants, etc., that scale the dependent variable.
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IMO adjustable parameters are those things that we vary in order to minimize a particular function in regression. In linear regression, the adjustable parameters are the slope and y-intercept. I am not sure that everyone likes this terminology or thinks that it is best.
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I usually call them fit parameters, whether the fit is linear or not.