Chapter 8: Problem 54
The following reaction is first order: \(\mathrm{C}_{2} \mathrm{H}_{6} \rightarrow 2 \mathrm{CH}_{3}\). If the rate constant is equal to \(5.5\) \(\times 10^{-4} \mathrm{~s}^{-1}\) at \(1000 \mathrm{~K}\), how long will it take for \(0.35\) mol of \(\mathrm{C}_{2} \mathrm{H}_{6}\) in a \(1.00 \mathrm{~L}\) container to decrease to \(0.10 \mathrm{~mol}\) in the same container? a. \(38 \mathrm{~min}\) b. \(26 \mathrm{~min}\) c. \(19 \mathrm{~min}\) d. \(68 \mathrm{~min}\)
Short Answer
Step by step solution
Identify the formula
Convert amounts to concentrations
Substitute the known values into the equation
Calculate the natural logarithm
Calculate time
Choose the correct option
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Rate Constant
The smaller the rate constant, the slower the reaction. Conversely, a higher rate constant leads to a faster reaction. In the given exercise, the rate constant is \( 5.5 \times 10^{-4} \mathrm{~s}^{-1} \). This tells us how the concentration of the reactant decreases over time in a first-order reaction.
- The rate constant remains constant only under steady conditions, like constant temperature.
- It's unique to every reaction and can vary with changes in temperature or the presence of a catalyst.
Natural Logarithm
In the formula \( t = \frac{1}{k} \ln \left( \frac{[A]_0}{[A]} \right) \), the natural logarithm converts the ratio of concentrations into a linear scale, which allows us to calculate the time needed for the reaction.
- Natural logarithms are expressed as powers of Euler's number \( e \), which is approximately 2.718.
- They play a crucial role in dealing with exponential growth or decay models, like the decrease of reactants in a chemical reaction.
Time Calculation for Reactions
For instance, substituting the given values \( k = 5.5 \times 10^{-4} \mathrm{~s}^{-1} \), \( [A]_0 = 0.35 \) mol/L, and \( [A] = 0.10 \) mol/L into the equation gives us the reaction time in seconds.
- This calculation is significant because it allows scientists to control and optimize reactions in various industries.
- Finding the precise time for reactions helps in managing resources and predicting the outcomes in production settings.
Concentration Conversions
For the exercise, we consider the ethanol molecule concentration change, from an initial \[ [A]_0 = 0.35 \text{ mol/L} \] to \[ [A] = 0.10 \text{ mol/L} \] within a 1.00 L container. Recognizing these concentrations correctly allows us to apply the kinetics equations accurately.
- Moles are a standard unit in chemistry expressing the amount of a chemical substance.
- Converting quantities to concentrations helps in directly applying them to kinetic formulas, especially when assessing reaction rates over a period.