David A. Tanzer, August 17, 2020
Analysis of reaction rates: first approach
Now let’s work out the equation for how the popping evolves over time.
Let Rate(t) be the average rate at which popping takes place at time $t$. Once the popping has begun, we expect Rate(t) to be high, and then dwindle down towards zero as more and more of the kernels have popped.
Suppose that Rate(20) = 5, which means that at 20 seconds in, the average rate popping is 5 kernels per second. Then, at that time, Popped is increasing (on the average) at 5 per second, and Unpopped is decreasing by 5 per second; these are the derivatives.
The derivative Popped'(20) is 5 kernels per second. Since Unpopped is always decreasing, its derivative is negative: Unpopped'(20) = -5 kernels per second.
Generalizing from t=20 to all times, we that:
- Unpopped'(t) = -Rate(t)
But what is Rate(t) itself?
The expected rate of popping
There are two factors involved in the expected rate of popping.
First, there are properties of the kernels themselves and how they respond to heat. We might imagine that some varieties of corn are slower cooking than others, due to differences in the physical structure of the kernels.
We’ll summarize all of these properties by a single number RC, called the rate coeffcient, which specifies the propensity of the kernels to pop. We might imagine two varieties of kernels, A and B, with different rate coefficients; so if RC(A) = 1 and RC(B) = 1.5, then all else being equal, B kernels will pop at 1.5 times the rate of A kernels.
The other factor affecting the rate is simply the number of kernels that are available in the state Unpopped at any moment: the rate of expected pops per second will be proportional to the number of unpopped kernels.
- Rate = RateCoefficient * Unpopped(t)
The rate equation
Let’s put our two equations together:
- Unpopped'(t) = -Rate(t)
- Rate(t) = RateCoefficient * Unpopped(t)
Putting these together, we get a single equation:
- Unpopped'(t) = – RateCoefficient * Unpopped(t)
This is called the rate equation; it is a differential equation, as it involves a derivative Unpopped'(t).
A fine point
With real popcorn, the rate coefficient, which describes propensities of the kernels to pop is not a constant, but depends on time: the longer it has cooked, the higher will be the propensity to pop. Indeed, even with a large population of kernels, at the beginning none will be popping. So a more accurate equation would be:
- Unpopped'(t) = – RateCoefficient(t) * Unpopped(t)
However, we’ve got enough on our hands analyzing systems even without the complications of varying rate coefficients.
Next time we will make a small digression, to look at the solution to a rate equation.
Copyright © 2020, David A. Tanzer. All Rights Reserved.