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Chapter 11: Problem 49

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?

Short Answer

Expert verified
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 by step solution

01

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.
02

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, \(A_t/A_0 = 0.55\).
03

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.
04

Write down the half-life equation for first-order reactions

The half-life equation for first-order reactions is: \[t_{1/2} = \frac{0.693}{k}\]
05

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.

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Most popular questions from this chapter

The thiosulfate ion \(\left(S_{2} O_{3}^{2-}\right)\) is oxidized by iodine as follows: $$2 \mathrm{S}_{2} \mathrm{O}_{3}^{2-}(a q)+\mathrm{I}_{2}(a q) \longrightarrow \mathrm{S}_{4} \mathrm{O}_{6}^{2-}(a q)+2 \mathrm{I}^{-}(a q)$$ In a certain experiment, \(7.05 \times 10^{-3} \mathrm{mol} / \mathrm{L}\) of \(\mathrm{S}_{2} \mathrm{O}_{3}^{2-}\) is consumed in the first 11.0 seconds of the reaction. Calculate the rate of consumption of \(\mathrm{S}_{2} \mathrm{O}_{3}^{2-} .\) Calculate the rate of production of iodide ion.

Sulfuryl chloride \(\left(\mathrm{SO}_{2} \mathrm{Cl}_{2}\right)\) decomposes to sulfur dioxide \(\left(\mathrm{SO}_{2}\right)\) and chlorine \(\left(\mathrm{Cl}_{2}\right)\) by reaction in the gas phase. The following pressure data were obtained when a sample containing \(5.00 \times 10^{-2}\) mol sulfury 1 chloride was heated to \(600 . \mathrm{K}\) in a \(5.00 \times 10^{-1}-\mathrm{L}\) container. Defining the rate as $$-\frac{\Delta\left[\mathrm{SO}_{2} \mathrm{Cl}_{2}\right]}{\Delta t}$$ a. determine the value of the rate constant for the decomposition of sulfuryl chloride at \(600 .\) K. b. what is the half-life of the reaction? c. what fraction of the sulfuryl chloride remains after \(20.0 \mathrm{h} ?\)

The activation energy for the reaction $$\mathrm{NO}_{2}(g)+\mathrm{CO}(g) \longrightarrow \mathrm{NO}(g)+\mathrm{CO}_{2}(g)$$ is \(125 \mathrm{kJ} / \mathrm{mol},\) and \(\Delta E\) for the reaction is \(-216 \mathrm{kJ} / \mathrm{mol}\). What is the activation energy for the reverse reaction \(\left[\mathrm{NO}(g)+\mathrm{CO}_{2}(g) \longrightarrow \mathrm{NO}_{2}(g)+\mathrm{CO}(g)\right] ?\)

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: $$\mathrm{CCl}_{2} \mathrm{F}_{2}(g) \stackrel{h v}{\longrightarrow} \mathrm{CF}_{2} \mathrm{Cl}(g)+\mathrm{Cl}(g)$$ Chlorine atoms can act as a catalyst for the destruction of ozone. The activation energy for the reaction $$\mathrm{Cl}(g)+\mathrm{O}_{3}(g) \longrightarrow \mathrm{ClO}(g)+\mathrm{O}_{2}(g)$$is \(2.1 \mathrm{kJ} / \mathrm{mol} .\) Which is the more effective catalyst for the destruction of ozone, Cl or NO? (See Exercise 75.)

The type of rate law for a reaction, either the differential rate law or the integrated rate law, is usually determined by which data is easiest to collect. Explain.
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