If the initial concentration of AB is 0.210 M, and the reaction mixture initially contains no products, what are the concentrations of A and B after 75 s?
The reaction is in the second order: rate=k[AB]^2
The rate constant is k=5.4*10^2 M*s
I don't understand how to find A and B separately.
To find the concentrations of A and B separately after 75 seconds, you can use the given rate equation and the rate constant. Let's go step by step:
First, let's assign variables for the concentrations of A, B, and AB at different time points in the reaction:
- [A]0: Initial concentration of A
- [B]0: Initial concentration of B
- [AB]0: Initial concentration of AB
- [A]: Concentration of A at a specific time t
- [B]: Concentration of B at a specific time t
- [AB]: Concentration of AB at a specific time t
Given:
- [AB]0 = 0.210 M (Initial concentration of AB)
- [A]0 = [B]0 = 0 M (Initial concentration of A and B)
- k = 5.4 * 10^2 M*s (Rate constant)
Since the reaction is stated to be second-order, the rate equation can be written as:
rate = k[AB]^2
We need to use this rate equation to find the concentrations of A and B after 75 seconds. At the start of the reaction, the concentration of AB is 0.210 M, and the concentrations of A and B are both 0 M.
Now, considering that [A]0 = [B]0 = 0 M, we can assume that both A and B are formed at equal rates and have equal concentrations at any given time during the reaction.
Therefore, after 75 seconds, the concentrations of A and B would be equal. Let's call this concentration [A] or [B]:
[A] = [B] (after 75 seconds)
Now you can use the rate equation to find [AB] after 75 seconds:
rate = k[AB]^2
Since we are considering 75 seconds, the rate can be written as the change in [AB] per unit time:
rate = d[AB]/dt
We can integrate this equation to solve for [AB] at 75 seconds:
∫d[AB]/[AB]^2 = ∫k dt
This integration will give us the relation:
-1/[AB] = kt + C
Now, using the given information that [AB]0 = 0.210 M initially (at t = 0 seconds), we can calculate the value of C:
-1/[AB]0 = k(0) + C
C = -1/[AB]0
Now we can substitute the values of [AB]0 = 0.210 M, k = 5.4 * 10^2 M*s, and C into the equation:
-1/[AB] = (5.4 * 10^2 M*s)t - 1/[AB]0
Rearranging the equation to solve for [AB]:
1/[AB] = 1/[AB]0 - (5.4 * 10^2 M*s)t
Plugging in the values for [AB]0 = 0.210 M and t = 75 s:
1/[AB] = 1/0.210 M - (5.4 * 10^2 M*s)(75 s)
Now, you can solve for [AB]:
1/[AB] = 4.76 M - 4.05 * 10^4 M
1/[AB] = -4.05 * 10^4 M
[AB] = -1/(4.05 * 10^4 M) (Note: The negative sign in this case indicates that the reaction has gone to completion.)
Since [A] = [B] (after 75 seconds), the concentrations of A and B would be equal to [AB] after 75 seconds:
[A] = [B] = -1/(4.05 * 10^4 M)
Therefore, the concentrations of A and B after 75 seconds would be approximately -2.47 * 10^-5 M.