Decomposition of an organic compound A follows first order kinetics. Initial concentration of A is 2 M. If rate constant of the decomposition is 3.58 × 10^−6 min^-1 , calculate the rate after 3 hours.
To calculate the rate after 3 hours, we first need to determine the concentration of compound A after 3 hours using the first-order decay equation:
A(t) = A₀ * e^(-kt)
Where:
A(t) = concentration of compound A at time t
A₀ = initial concentration of compound A
k = rate constant
t = time
Given:
A₀ = 2 M (initial concentration)
k = 3.58 × 10^(-6) min^(-1) (rate constant)
t = 3 hours = 3 * 60 = 180 minutes (time in minutes)
Substituting the given values into the equation:
A(180) = 2 * e^(-3.58 × 10^(-6) * 180)
Calculating the value of A(180):
A(180) ≈ 2 * e^(-0.0006464) ≈ 2 * 0.99935362 ≈ 1.99870724 M
Therefore, the concentration of compound A after 3 hours is approximately 1.9987 M.
To calculate the rate of decomposition of an organic compound after a specific time, we can use the first-order reaction equation:
ln(A/A₀) = -kt
where:
- A is the concentration of the compound at time t
- A₀ is the initial concentration of the compound
- k is the rate constant
- t is the time
In this case, we need to calculate the rate of decomposition after 3 hours. The rate is given by the equation:
rate = -dA/dt
Let's calculate the concentration of the compound at 3 hours using the first-order equation:
ln(A/A₀) = -kt
ln(A/2) = -(3.58 × 10^(-6) min^(-1)) × (3 hours) × (60 min/hour)
ln(A/2) = -(3.58 × 10^(-6) min^(-1)) × (3 hours) × (60 min/hour)
ln(A/2) = -(3.58 × 10^(-6)) × (180)
Next, we solve for A:
A/2 = e^-(3.58 × 10^(-6) × 180)
A = 2 × e^-(3.58 × 10^(-6) × 180)
Now, we can calculate the rate at 3 hours using the rate equation:
rate = -dA/dt
rate = -d[2 × e^-(3.58 × 10^(-6) × 180)]/dt
To find the rate at 3 hours, we can differentiate the concentration equation with respect to time, t, and substitute t = 3 hours. However, since we don't have any information about the mechanism or intermediates, it is not possible to calculate the rate at 3 hours without knowing additional details.
Alternatively, if the half-life of the reaction is provided, we can calculate the rate at 3 hours using the equation:
rate = k × A
where A is the remaining concentration of the compound after 3 hours, which can be calculated using the half-life:
t₁/2 = (ln2) / k
Substituting the given rate constant (k) into the half-life equation, we can calculate the half-life (t₁/2). If we have the half-life, we can use it to find the remaining concentration (A) at 3 hours and calculate the rate.