Popcorn kernals pop independently (i.e. unimolecularly). For one brand at constant temperature, 11 kernals pop in 10 seconds when 240 kernals are present. After 145 kernals have popped, how many kernals will pop in 10 seconds?(Your answer may include fractions of a kernal)
I understand that rate constants and rates are involved with the problem, but I'm not sure how to approach it..
I believe you want to use
ln(No/N) = kt.
Substitute No = 240
N = 240-11 = ??
t = 10 seconds.
Solve for k.
Then use k in
ln(No/N) = kt
No = 240-145 = ??
N = unknown
you know k and t. Solve for N, which will be number of kernals remaining; therefore, No-N must be the amount popped. Check my thinking.
To approach this problem, we can use the concept of reaction rates and the rate constant. Let's break it down step by step:
1. Start by determining the rate at which the popcorn kernels are popping. We are given that 11 kernels pop in 10 seconds when there are 240 kernels present. So the overall rate of popping is 11 kernels / 10 seconds.
2. Now, let's calculate the rate constant. The rate constant (k) describes the speed at which the reaction occurs. In this case, it represents the rate at which individual kernels pop. To find the rate constant, divide the rate by the number of kernels present:
k = rate / number of kernels
k = 11 kernels / 10 seconds / 240 kernels
k = 0.0458 seconds^(-1)
3. Now, let's determine how many kernels will pop after 145 kernels have already popped. Since the kernels pop independently, we can use the exponential decay formula:
N(t) = N(0) * e^(-kt)
- N(t) represents the number of kernels remaining after time t.
- N(0) is the initial number of kernels.
- k is the rate constant.
- t is the time.
In this case, N(0) is 240 kernels, and t is 10 seconds. We want to find N(t) when 145 kernels have already popped. So we substitute these values into the formula:
N(t) = 240 * e^(-0.0458 * 10)
4. Calculate N(t) using a calculator or computer:
N(t) ≈ 240 * 0.3322
N(t) ≈ 79.73 kernels
Therefore, approximately 79.73 kernels will pop in the next 10 seconds after 145 kernels have already popped.
To solve this problem, we can use the concept of reaction rates. The reaction rate is defined as the change in the number of reactants or products with respect to time.
Given that 11 kernels pop in 10 seconds when there are 240 kernels present, we can determine the reaction rate constant (k) as follows:
Rate = k * [kernels]
11 kernels / 10 seconds = k * 240 kernels
Simplifying the equation, we find:
k = (11 kernels / 10 seconds) / (240 kernels) = 0.0458 seconds^(-1)
Now, we can use this rate constant to determine the number of kernels that will pop in the next 10 seconds, after 145 kernels have already popped.
Let's denote the number of kernels that will pop in the next 10 seconds as x.
Using the rate equation:
Rate = k * [kernels]
x kernels / 10 seconds = 0.0458 seconds^(-1) * (240 kernels - 145 kernels)
Simplifying the equation, we find:
x / 10 seconds = 0.0458 seconds^(-1) * 95 kernels
x = (0.0458 seconds^(-1) * 95 kernels) * 10 seconds
x ≈ 43 kernels
Therefore, approximately 43 kernels will pop in the next 10 seconds after 145 kernels have already popped.