If the half-life of a first-order decomposition reaction is 24.2 minutes, how long will it take for 10.0 moles of the reactant to decompose so that only 2.5 mol remains?
k = 0.693/t1/2
ln(No/N) = kt
No = 10.0
N = 2.50
K from above.
Solve for t in minutes.
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36.3
To solve this problem, we can use the mathematical expression for a first-order decomposition reaction:
N(t) = N₀ * e^(-kt)
Where:
- N(t) is the remaining number of moles at time t
- N₀ is the initial number of moles
- k is the rate constant
- t is time
First, we need to find the rate constant (k) using the information given. The half-life (t₁/₂) of a first-order reaction can be related to the rate constant as follows:
t₁/₂ = 0.693 / k
Rearranging the equation, we find:
k = 0.693 / t₁/₂
Substituting t₁/₂ = 24.2 minutes:
k = 0.693 / 24.2
Now, we can use the equation N(t) = N₀ * e^(-kt) to find the time it takes for 10.0 moles of the reactant to decompose to 2.5 moles:
2.5 = 10.0 * e^(-kt)
Dividing both sides by 10.0:
0.25 = e^(-kt)
Taking the natural logarithm of both sides:
ln(0.25) = -kt
Now we can solve for t:
t = -ln(0.25) / k
Substituting the value of k we calculated earlier:
t = -ln(0.25) / (0.693 / 24.2)
Evaluating this expression will give us the time it takes for the reaction to decompose from 10.0 moles to 2.5 moles.