The half-life of a first order reaction is determined to be 74.5 years. How long will it take for the concentration of the reactant to reach 2% of its initial value?
420 years
74.5years X (multiply) with 5.64386
To determine how long it will take for the concentration of the reactant to reach 2% of its initial value, we can use the mathematical equation for a first-order reaction:
ln(C/C₀) = -kt
Where:
C is the concentration at a given time,
C₀ is the initial concentration,
k is the rate constant, and
t is the time.
We know that the half-life of the reaction is 74.5 years. The half-life duration represents the time it takes for the concentration to decrease by half, which can be expressed mathematically as:
0.5 = e^(-k * 74.5)
To solve for k, we need to isolate it. Taking the natural logarithm (ln) of both sides gives us:
ln(0.5) = -k * 74.5
Now, we can calculate the value of k by substituting the natural logarithm of 0.5:
k = ln(0.5) / -74.5
Next, we can determine how long it will take for the concentration to reach 2% of its initial value. Considering that 2% is equivalent to 0.02 times the initial concentration, we need to solve for t when C/C₀ equals 0.02:
ln(0.02) = -k * t
Substituting the value of k we obtained earlier:
ln(0.02) = (ln(0.5) / -74.5) * t
Now, we can solve for t by isolating it:
t = ln(0.02) / (ln(0.5) / -74.5)
Evaluating this expression will provide the time it takes for the concentration to reach 2% of its initial value.