Understanding Half-Life in First-Order Reactions

If the rate constant for a first order reaction is 0.693 min^-1, what is the half-life of this reaction?

• A. 0.500 min
• B. 1.000 min
• C. 2.000 min
• D. 3.464 min

Answer: (B)

In a first-order reaction, the relationship between the rate constant and the half-life is particularly straightforward. The half-life (t₁/₂) of a first-order reaction is given by the formula: \[ t_{1/2} = \frac{0.693}{k} \] where k is the rate constant. Given that the rate constant (k) is 0.693 min⁻¹, you can substitute this value into the equation to find the half-life: \[ t_{1/2} = \frac{0.693}{0.693 \, \text{min}^{-1}} \] When you perform the calculation, the 0.693 in the numerator and denominator cancel each other out, leading to: \[ t_{1/2} = 1.000 \, \text{min} \] This result indicates that the half-life of the reaction is 1.000 min, which embodies the characteristic of first-order kinetics where the half-life remains constant regardless of the initial concentration of the reactants.

When it comes to chemical reactions, half-life is an absolute cornerstone concept. It's like a ticking clock, marking the time it takes for half of a substance to undergo transformation. Now, if you're gearing up for the USA Biology Olympiad (USABO), getting a grip on first-order reactions is essential—and here's the kicker: it’s not as daunting as it seems.

Let’s break it down. Imagine you have a first-order reaction, and the rate constant (k) is 0.693 min⁻¹. You might be asking, “Alright, how does this relate to half-life?” Well, here's the thing: for first-order reactions, the half-life (t₁/₂) is a nifty little formula:

[ t_{1/2} = \frac{0.693}{k} ]

Don’t worry; it’s easier than your neighbor’s dog winking at you. Just plug in that rate constant value. So, if k = 0.693 min⁻¹, the calculation becomes:

[ t_{1/2} = \frac{0.693}{0.693} ]

And surprise, surprise—the numbers cancel! What you’re left with is:

[ t_{1/2} = 1.000 , \text{min} ]

So, the half-life of this reaction is 1 minute. Simple, right? This consistency in half-life regardless of concentration is what sets first-order reactions apart. It’s like they have a built-in timer that never changes!

But why does this matter, beyond just crunching numbers? Understanding half-life isn’t just a lab exercise; it reflects real biological processes. Think about it: drug metabolism in our bodies often follows first-order kinetics. The body breaks down substances at a steady rate, impacting everything from medications to nutrients. So, grasping this concept is vital for anyone entering the biological realm, especially competitive environments like the USABO.

Now, you may wonder how this applies to biology on a broader scale. Let’s take a little side trip. In practices like pharmacology or toxicology, knowing how long it takes for half a drug to be eliminated can be a game-changer for treatment plans. If a doctor knows that a patient metabolizes a certain medication quickly or slowly, they can adjust dosages accordingly. It’s practical knowledge that translates directly into patient care.

In essence, while chemistry might seem like just another hurdle in your studies, concepts like half-life are useful tools for understanding the living world. And it keeps getting better! As you continue preparing for the USABO, remember that mastering these core ideas not only helps you ace exams but also enriches your appreciation for biological interactions.

So, as you move forward in your studies, keep half-life in your mental toolkit. It’s not just about getting the right answer on a practice exam; it’s about weaving together the fundamental patterns of life, one reaction at a time! And who knows, maybe one day, you’ll be the one explaining these concepts to the next generation of eager learners.