Problem 49 - Chemistry An Atoms First Approach

Question

A certain first-order reaction is \(45.0 \%\) complete in 65 s. What are the values of the rate constant and the half-life for this process?

Step-by-Step Solution

Answer

The rate constant (k) for this first-order reaction is approximately \(0.00966 \, s^{-1}\), and its half-life is approximately 71.7 seconds.

Step 1: Write down the first-order integrated rate law equation

The first-order integrated rate law equation is: \[\ln\left(\frac{A_t}{A_0}\right) = -kt\]Where \(A_t\) is the amount of reactant remaining at time t, \(A_0\) is the initial amount of reactant, k is the rate constant, and t is the time.

Step 2: Determine the fraction of reactant remaining

The problem states the reaction is 45% complete in 65 seconds. Therefore, we need to find the fraction of the reactant remaining (A_t/A_0). Since 45% of the reactant is consumed, 55% of the reactant remains. So, \(A_t/A_0 = 0.55\).

Step 3: Plug in the values and solve for the rate constant, k

Now we have the equation: \[\ln\left(\frac{A_t}{A_0}\right) = -kt\]Plug in the known values to the equation: \[\ln(0.55) = -k(65)\] Now, solve for k: \[k = -\frac{\ln(0.55)}{65}≈0.00966 \, s^{-1}\]So, the rate constant k is approximately 0.00966/s.

Step 4: Write down the half-life equation for first-order reactions

The half-life equation for first-order reactions is: \[t_{1/2} = \frac{0.693}{k}\]

Step 5: Plug in the value of k to find the half-life

Now, plug in the value of k in the half-life equation: \[t_{1/2} = \frac{0.693}{0.00966} ≈ 71.7 \, s\] So, the half-life for this first-order reaction is approximately 71.7 seconds In conclusion, the rate constant for this first-order reaction is approximately 0.00966/s, and its half-life is approximately 71.7 seconds.

Key Concepts

Rate Constant

In chemical kinetics, the rate constant (denoted as \(k\)) is a crucial value that tells us how fast a reaction proceeds. For first-order reactions, the rate constant directly relates to the speed at which the reactant transforms into the product. This value remains constant as long as the temperature is unchanged.
A higher rate constant implies a faster reaction, while a lower rate constant means the reaction is slower.

• The unit for the rate constant in a first-order reaction is \(s^{-1}\).
• The value of the rate constant is typically determined using the integrated rate law equation.

Knowing the rate constant is essential for predicting reaction behavior and calculating other key metrics such as the half-life.

Half-life

The half-life of a reaction is the time required for half of the reactant to be consumed in a chemical reaction. In first-order reactions, the half-life is independent of the initial concentration of the reactant.

• For a first-order reaction, the half-life can be calculated using the formula: \[ t_{1/2} = \frac{0.693}{k} \]
• This formula shows that the half-life is inversely proportional to the rate constant \(k\).

In simple terms, if the rate constant is large, the half-life is short, indicating that the reaction proceeds rapidly. Understanding half-life is important in contexts such as pharmaceuticals and radioactive decay, where it helps determine how long a substance will remain in a system.

Integrated Rate Law

The integrated rate law for first-order reactions is a mathematical expression that relates the concentration of the reactant to time. For a first-order reaction, it is expressed as:\[ \ln\left(\frac{A_t}{A_0}\right) = -kt \]
This equation allows us to calculate the concentration of the reactant at any given time.

• \(A_t\) is the concentration at time \(t\).
• \(A_0\) is the initial concentration.
• \(k\) is the rate constant.
• \(t\) is the time elapsed.

By rearranging the formula, you can solve for any one variable if the others are known. The integrated rate law helps chemists understand the kinetics of a reaction and predict future concentrations of reactants over time.

Chemical Kinetics

Chemical kinetics is the area of chemistry concerned with the speeds of chemical reactions and the factors affecting these speeds. Essential concepts in chemical kinetics include the rate constant, reaction order, and the effect of temperature on reaction rates.

• Understanding kinetics helps us control reactions, optimize conditions, and design reactors for industrial applications.
• Kinetics provides insights into reaction mechanisms, the step-by-step processes by which reactions occur.
• First-order kinetics applies when the reaction rate depends linearly on the concentration of a single reactant.

By studying chemical kinetics, scientists gain valuable information about potential reaction pathways and how changes in conditions like temperature and concentration can affect reaction rates.