The half-life of a first order reaction is determined to be 60.5 years. How long will it take for the concentration of the reactant to reach 2% of its initial value?
To determine how long it will take for the concentration of the reactant to reach 2% of its initial value in a first-order reaction, we need to use the half-life and the equation for a first-order reaction.
The half-life of a first-order reaction is the time it takes for the concentration of the reactant to decrease to half its initial value. In this case, the half-life is given as 60.5 years.
The equation for the concentration of a reactant in a first-order reaction is:
C(t) = C₀ * e^(-kt)
where C(t) is the concentration at time t, C₀ is the initial concentration, k is the rate constant, and e is Euler's number (approximately 2.71828).
Since we know the half-life, we can use it to find the rate constant, k:
0.5 = e^(-k * 60.5)
To solve for k, we take the natural logarithm (ln) of both sides:
ln(0.5) = -k * 60.5
Divide both sides by -60.5:
k = ln(0.5) / -60.5
With the rate constant k known, we can determine the time it takes for the concentration to reach 2% of its initial value by substituting the appropriate values into the equation:
0.02 = C₀ * e^(-kt)
Since we want to solve for time (t), we can rearrange the equation as:
t = (ln(0.02) - ln(C₀)) / -k
Now you can substitute the known values into the equation and calculate the time it will take for the concentration to reach 2% of its initial value.