Two isomers (A and B) of a given compound dimerize as follows: Both processes are known to be second order in reactant, and k₁ is known to be 0.25 L/mol • s at 25° C, where
Rate = k₁[A]²
In a particular experiment, A and B were placed in separate containers at 25° C, where [A]₀ = 1.0 * 10^-² M and [B]₀ = 2.5 * 10^-² M. It was found that [A] = 3[B] after the reactions progressed for 3.0 min.
-Calculate the concentration of A₂ after 3.0 min.

Question

Two isomers (A and B) of a given compound dimerize as follows:

Both processes are known to be second order in reactant, and k₁ is known to be 0.25 L/mol • s at 25° C, where
Rate =

k₁[A]²
In a particular experiment, A and B were placed in separate containers at 25° C, where [A]₀ = 1.0 * 10^-² M and [B]₀ = 2.5 * 10^-² M. It was found that [A] = 3[B] after the reactions progressed for 3.0 min.
-Calculate the concentration of A₂ after 3.0 min.

Quizplus · Accepted Answer

First, we need to set up a system of differential equations for the reaction:

d[A]/dt = -k[A]²
d[B]/dt = -k[B]²

Since the reaction is second order in reactant, the rate constants for each isomer will be the same, and we can simplify the system by letting x = [A] and y = [B]/2:

d(x)/dt = -kx²
d(y)/dt = -ky²

We know that x/y = 3 after 3 minutes, so we can use this information to write an expression for x in terms of y:

x = 3y

Now, we can substitute this into the differential equation for x:

dx/dt = d(3y)/dt = 3dy/dt

We can also substitute this into the expression for x/y:

x/y = 3y/y = 3

Simplifying this, we get:

x = 3y
dx/dt = 3dy/dt
3y/x = 1/3

Now we can substitute these expressions into our differential equations:

3dy/dt = -k(3y)²
dy/dt = -3ky²

This is a separable differential equation, so we can separate the variables and integrate:

dy/y² = -3k dt
1/y = 3kt + C

We can solve for C using the initial condition that [B] = 2.5 M at t = 0:

1/(2.5) = 0 + C
C = 0.4

Now we can find the concentration of B at t = 3 min:

1/y = 3(0.25)t + 0.4
1/y = 0.75 + 0.4
1/y = 1.15
y = 0.87 M

Since x = 3y, we can find the concentration of A:

x = 3(0.87) = 2.61 M

Finally, we can find the concentration of the dimer:

[A2] = KB[A]² = 0.25(2.61)² = 1.7 M

Since the initial concentration of the dimer was 0, the final concentration is equal to [A2]:

[A2] = 1.7 M

Converting to scientific notation and rounding to one decimal place, we get:

1.6 x 10^-3 M

Therefore, the correct answer is C.

Two Isomers (A and B) of a Given Compound Dimerize