Treat everything on the RHS as a constant in time.
You're gonna perform the integrals, which is quite straight-forward, and then you'll have to define such that you get the equation you need. Can you do the next step?
I'm a bit rusty on my calc, and hope someone here can help me. In reviewing a particular scientific paper, the reaction rate equation is expressed as:
Where is the reduction in property with respect to time
A is the steric constant
k is the gas constant or Bolzmann constant (depending on units)
T is absolute temperature
E is the activation energy of the reaction
The paper then says "Integration of the rate equation, followed by the taking of logarithms, results in an equation of the form"
I understand this to be true, but I am having a hard time getting from the first equation to the second equation on my own. If anyone could break it down step by step for me, I would be most appreciative.
Thanks for the quick response...OK, So I get
and if I solve for t I get:
I guess I need to assume R is a constant. If R is a constant, then R+C1/A is a constant I can call D
Am I on the right track?
I get stuck If I take the natural log of both sides of this
Because of the C2 term.
If I ignore the C2 term, then it seems to work out
If I call lnD=B, then I get my desired form
So, please help me see where I am going wrong. How can I ignore the constant created by the integration of dt? Or am I missing something?
Thank you again.
You don't need two constants of integration - one will suffice. I did it this way:
I moved the constant of integration over to the other side, and then took the logarithm, etc.
is most definitely not a constant. If it were, its derivative would be zero, not the expression we have for When you define your , it's obviously going to have in it somewhere. Why does have to be constant? Does the paper say is constant?
OK...I think I'm starting to see the light. You are right, R is not a constant, but a function of T. Certainly it needs to be accounted for. And how quickly I forgot that we are treating the initial RHS as a constant, and a constant times a constant is...duh.
And the paper doesn't say explicitly that B is a constant, but the formula we arrive at looks like the general form for a line if we substitute m=E/k and, x=1/T, and y=lnt, so I was thinking B would be constant as an intercept.
You are right, R is a function of t (time). dR/dt is a function of T (Temperature).
But the paper does treat the equation like a straight line, as it then says "A plot of lnt versus 1/T is then a straight line of slope E/k, and is known as the Arrhenius plot."
The plots being generated in this paper plot experimental data of lifetime (t) for multiple Temperatures (T). ln t is used for the y axis and 1/T is used for the x axis, then a best fit line is drawn, and the slope of that line is used to calculate E (Since k is a constant)
But the value of R used to determine lifetime is being defined in the experiment...Ah Ha! So if we are saying that we are going to plot these points for values of t associated with a specific value of R, we will, for the purposes of the plot, be holding R constant. R varies with time, but the value of time we are using for any given T corresponds with a set, consistent value of R. So, in essence, we are allowing t to vary for a given T to get a specific value of R. It appears that this is why we get a straight line.
Thank you so much for helping me to work through this.