N2O undergoes thermal decomposition at 730°C to nitrogen (N2) and oxygen (O2) via first-order kinetics with a half-life of 3.58 x 103 minutes. Calculate the time required for 90% of a sample of N2O to decompose at 730°C.
I think this is the easiest way.
k = 0.693/t1/2 and substitute k into the following:
ln(No/N) = kt.
I would call No = 100, then
N = 10 (if 90% is gone)
k from above
t = solve for this. The unit will be the unit used for the half life to solve for k.
i am so sad
To calculate the time required for 90% of a sample of N2O to decompose, we need to determine the reaction rate constant (k) and then use it in the integrated rate equation of a first-order reaction to solve for the time.
First, let's find the reaction rate constant (k) using the half-life (t1/2) value provided.
The half-life of a first-order reaction is related to the rate constant (k) by the following equation:
t1/2 = (0.693 / k)
Rearranging the equation to solve for k:
k = 0.693 / t1/2
Substituting the given half-life value:
k = 0.693 / (3.58 x 10^3 minutes)
Calculating the value of k:
k ≈ 1.94 x 10^-4 min^-1
Now that we have the rate constant (k), we can use the integrated rate equation of a first-order reaction to solve for the time required for 90% decomposition.
The integrated rate equation is given by:
ln([A]t / [A]0) = -kt
Where:
[A]t = concentration of N2O at time t
[A]0 = initial concentration of N2O
k = rate constant
t = time
Since we want to find the time required for 90% decomposition, we can rearrange the equation as follows:
ln([A]t / [A]0) = -kt
ln(0.1 / 1) = -k * t
Substituting the known values:
ln(0.1) = -(1.94 x 10^-4 min^-1) * t
Now, solving for t using logarithmic properties:
t = -ln(0.1) / (k)
Calculating the value of t:
t ≈ -ln(0.1) / (1.94 x 10^-4 min^-1)
t ≈ 210.85 minutes
Therefore, it would take approximately 210.85 minutes for 90% of the sample of N2O to decompose at 730°C.