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Chapter 56: Problem 2

A student obtains data for a first-order reaction at a given temperature, and then makes a graph of \(\ln (\mathrm{R})\) (along the vertical axis) versus \(t\) (along the horizontal axis). She notes that the resulting plot appears to correspond to a straight-line relationship. The student then determines the slope and intercept of the best-fit straight line. Describe how the student could use the slope and/or intercept of the best-fit straight line to determine: a) the rate constant for the reaction at the given temperature. b) the value of \((\mathrm{R})_{0}\)

Short Answer

Expert verified
The rate constant for the reaction at the given temperature can be obtained by taking the absolute value of the slope of the plot \(\ln R\) vs \(t\). The value of \(R_0\) can be calculated by exponentiating the y-intercept of this plot.

Step by step solution

01

Identify the Slope

Determine the slope of the best-fit straight line. This slope, when measured, represents \(-k\), the negative of the rate constant, in the linear form of the first-order rate equation.
02

Calculate the Rate Constant

The rate constant can be found by taking the absolute value of the slope determined in Step 1. This means we have \(k = \left| slope \right|\).
03

Identify the Intercept

Determine the y-intercept of the best-fit straight line. This intercept is equivalent to \(\ln(R_0)\) in the linear form of the first-order rate equation.
04

Calculate the Initial Reaction Rate

The initial reaction rate, \(R_0\), can be calculated by taking the exponent of the intercept. This would be done by \((R)_{0} = e^{intercept}\).

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Most popular questions from this chapter

A student obtains data for a second-order reaction at a given temperature, and then makes a graph of \(1 /(\mathrm{R})\) (along the vertical axis) versus \(t\) (along the horizontal axis). He notes that the resulting plot appears to correspond to a straight-line relationship. The student then determines the slope and intercept of the best-fit straight line. Describe how the student could use the slope and/or intercept of the best-fit straight line to determine: a) the rate constant for the reaction at the given temperature. b) the value of \((\mathrm{R})_{\mathrm{o}}\)

Recall that for a first-order reaction: $$ \ln (\mathrm{R})=\ln (\mathrm{R})_{0}-k t $$ a) When \(t=t_{1 / 2}\), what is the value of \((\mathrm{R})\) in terms of \((\mathrm{R})_{0}\) ? b) Show that \(t_{1 / 2}=\frac{\ln 2}{k}=\frac{0.693}{k}\) for a first-order reaction.

Recall that for a second-order reaction: $$ \frac{1}{(\mathrm{R})}=\frac{1}{(\mathrm{R})_{\mathrm{o}}}+k t $$ a) When \(t=t_{1 / 2}\), what is the value of \((\mathrm{R})\) in terms of \((\mathrm{R})_{0}\) ? b) Show that \(t_{1 / 2}=\frac{1}{k(\mathrm{R})_{\mathrm{o}}}\) for a second- order reaction.
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