The half-life of a first order reaction is determined to be 72.5 years. How long will it take for the concentration of the reactant to reach 2% of its initial value?
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To determine how long it will take for the concentration of the reactant to reach 2% of its initial value, we need to use the concept of half-life.
In a first-order reaction, the half-life is defined as the time it takes for the concentration of the reactant to decrease to half of its initial value. We are given that the half-life of the reaction is 72.5 years.
We can use the following formula to calculate the time it takes for the concentration to reach a specific percentage of its initial value:
t = (ln(C₀/Cₙ))/k
where:
t = time
C₀ = initial concentration of reactant
Cₙ = final concentration of reactant (2% of its initial value)
k = rate constant
In this case, we know the half-life (72.5 years), but we need to determine the rate constant (k). The rate constant for a first-order reaction can be calculated using the formula:
k = ln(2)/t₁/₂
where:
t₁/₂ = half-life of the reaction.
Plugging in the values, we have:
k = ln(2)/72.5
Next, we can substitute the rate constant (k) into the formula for time (t):
t = (ln(C₀/Cₙ))/(ln(2)/72.5)
Since we want to find the time it takes for the concentration to reach 2% of its initial value, we can set Cₙ = 0.02 * C₀:
t = (ln(C₀/(0.02*C₀)))/(ln(2)/72.5)
Simplifying the expression:
t = (ln(1/0.02))/(ln(2)/72.5)
Finally, we can calculate t:
t ≈ (ln(50))/(ln(2)/72.5)
Using a calculator, we evaluate the natural logarithm of 50 and divide it by the natural logarithm of 2 divided by 72.5 to get the value of t.