The half-life for the zero order reaction A → Products is 276 s. What is the value of the rate constant for the reaction if the initial concentration of A is 0.789?
Answer (2)
Identify the half-life formula for a zero-order reaction: t 1/2 = 2 k [ A ] 0 .
Rearrange the formula to solve for the rate constant: k = 2 t 1/2 [ A ] 0 .
Substitute the given values [ A ] 0 = 0.789 and t 1/2 = 276 s into the equation.
Calculate the rate constant: k = 2 × 276 0.789 ≈ 0.00143 .
Explanation
Problem Setup We are given a zero-order reaction A → Products with a half-life t 1/2 = 276 s and an initial concentration [ A ] 0 = 0.789 . We need to find the rate constant k for this reaction.
Half-Life Formula For a zero-order reaction, the half-life is given by the formula:
t 1/2 = 2 k [ A ] 0
where t 1/2 is the half-life, [ A ] 0 is the initial concentration, and k is the rate constant.
Solving for k We can rearrange the formula to solve for k :
k = 2 t 1/2 [ A ] 0
Substitution Now, we substitute the given values into the equation:
k = 2 × 276 0.789
k = 552 0.789
Calculation Calculating the value of k :
k ≈ 0.0014293478260869565
Rounding to three significant figures, we get:
k ≈ 0.00143
Final Answer Therefore, the rate constant for the reaction is approximately 0.00143 .
Examples
Consider a scenario where a drug degrades via a zero-order reaction. Knowing the half-life and initial concentration, we can determine the rate at which the drug degrades. This is crucial in pharmacology to ensure the drug maintains its therapeutic efficacy over its shelf life. For instance, if a drug's initial concentration is 0.789 M and its half-life is 276 seconds, we can calculate the rate constant to predict how quickly the drug concentration will decrease over time, ensuring proper dosing and effectiveness.
The rate constant k for the zero-order reaction is calculated using the half-life formula as k ≈ 0.00143 M/s . This is derived from the relationship between half-life and initial concentration specifically for zero-order reactions. By substituting the given values into the formula, we find the rate constant accurately.
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