Answer :
Final answer:
To calculate the rate constant (k) for the first-order reaction, the integrated first-order rate law in logarithmic form is used, where ln(0.645) = -k(4.90 min) to solve for k.
Explanation:
The student is asking to calculate the rate constant of a first-order reaction that is 35.5 percent complete after 4.90 minutes at a temperature of 25°C. To solve for the rate constant (k), we can use the integrated first-order rate law:
[ A ] = [ A ]_0 * e-kt
where [ A ] is the concentration of reactants remaining, [ A ]_0 is the initial concentration of reactants, k is the rate constant, and t is the time. Since the reaction is 35.5% complete, 100% - 35.5% = 64.5% of the reactant remains. Converting this to a decimal gives us 0.645.
We assume the initial concentration [ A ]_0 to be 1 for simplicity (since concentrations cancel out when calculating percent completion). The reaction is 35.5 percent complete, which implies that [ A ] = 0.645 * [ A ]_0. The first-order integrated rate law in its logarithmic form is:
ln([ A ]/[ A ]_0) = -kt
Substituting in our values:
ln(0.645) = -k(4.90 min)
Now solve for k:
k = -ln(0.645) / 4.90 min
After calculating, we obtain the value of the rate constant k. The unit of k will be in min-1 since the time given is in minutes. Note, the actual numerical value of the rate constant is not provided, as the goal is to show the methodology.
