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Chapter 12: Problem 45

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

Short Answer

Expert verified
The rate constant for this first-order reaction is approximately \(k \approx 0.01169 s^{-1}\). The half-life for this first-order reaction is approximately \(t_{1/2} \approx 59.27s\).

Step by step solution

01

Determine the reaction completion and time elapsed

We are given that the reaction is 45.0% complete in 65 seconds.
02

Use the formula for first-order reaction

We can use the first-order reaction formula: \[ln(\frac{A_0}{A_t}) = kt\] Where: - \(A_0\) is the initial concentration - \(A_t\) is the concentration at time t - \(k\) is the rate constant - \(t\) is the time elapsed Since the reaction is 45.0% complete, we have 55.0% remaining. Assuming the initial concentration is 1, we can say that \(A_t = 0.55\). Now, substitute the known values into the formula: \[ln(\frac{1}{0.55}) = k \cdot 65s\]
03

Solve for the rate constant, k

We rearrange the formula to solve for k: \[k = \frac{ln(\frac{1}{0.55})}{65s}\] Now calculate k: \[k \approx 0.01169 s^{-1}\] So the rate constant for this first-order reaction is approximately 0.01169 s^(-1).
04

Determine the half-life (t) for a first-order reaction

The half-life formula for a first-order reaction can be written as: \[t_{1/2} = \frac{ln(2)}{k}\] Now, substitute the calculated value of k into the formula: \[t_{1/2} = \frac{ln(2)}{0.01169s^{-1}}\]
05

Calculate the half-life

Finally, calculate the half-life: \[t_{1/2} \approx 59.27s\] So the half-life for this first-order reaction is approximately 59.27 seconds.

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Most popular questions from this chapter

One of the concerns about the use of Freons is that they will migrate to the upper atmosphere, where chlorine atoms can be generated by the following reaction: $$ \mathrm{CCl}_{2} \mathrm{~F}_{2} \stackrel{\mathrm{hv}}{\longrightarrow} \mathrm{CF}_{2} \mathrm{Cl}+\mathrm{Cl} $$ Chlorine atoms can act as a catalyst for the destruction of ozone. The activation energy for the reaction $$ \mathrm{Cl}+\mathrm{O}_{3} \longrightarrow \mathrm{ClO}+\mathrm{O}_{2} $$ is \(2.1 \mathrm{~kJ} / \mathrm{mol}\). Which is the more effective catalyst for the destruction of ozone, \(\mathrm{Cl}\) or \(\mathrm{NO}\) ? (See Exercise \(69 .\) )

The reaction \(\mathrm{H}_{2} \mathrm{SeO}_{3}(a q)+6 \mathrm{I}^{-}(a q)+4 \mathrm{H}^{+}(a q)\) \(\longrightarrow \mathrm{Se}(s)+2 \mathrm{I}_{3}^{-}(a q)+3 \mathrm{H}_{2} \mathrm{O}(l)\) was studied at \(0^{\circ} \mathrm{C}\), and the following data were obtained:

Chemists commonly use a rule of thumb that an increase of \(10 \mathrm{~K}\) in temperature doubles the rate of a reaction. What must the activation energy be for this statement to be true for a temperature increase from 25 to \(35^{\circ} \mathrm{C}\) ?

How does temperature affect k, the rate constant? Explain.

Consider the following statements: "In general, the rate of a chemical reaction increases a bit at first because it takes a while for the reaction to get 'warmed up.' After that, however, the rate of the reaction decreases because its rate is dependent on the concentrations of the reactants, and these are decreasing." Indicate everything that is correct in these statements, and indicate everything that is incorrect. Correct the incorrect statements and explain.
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