91Ó°ÊÓ

The reaction \(A \rightarrow B\) is carried out in a laboratory reactor. According to a published article the concentration of A should vary with time as follows: \(C_{\mathrm{A}}=C_{\mathrm{A} 0} \exp (-k t)\) where \(C_{\mathrm{A} 0}\) is the initial concentration of \(\mathrm{A}\) in the reactor and \(k\) is a constant. (a) If \(C_{\mathrm{A}}\) and \(C_{\mathrm{A} 0}\) are in \(\mathrm{Ib}-\) moles \(/ \mathrm{ft}^{3}\) and \(t\) is in minutes, what are the units of \(k ?\) (b) The following data are taken for \(C_{\mathrm{A}}(t):\) $$\begin{array}{cc}\hline t(\min ) & C_{\mathrm{A}}\left(\mathrm{lb}-\mathrm{mole} / \mathrm{ft}^{3}\right) \\\\\hline 0.5 & 1.02 \\\1.0 & 0.84 \\\1.5 & 0.69 \\\2.0 & 0.56 \\\3.0 & 0.38 \\\ 5.0 & 0.17 \\\10.0 & 0.02 \\\\\hline\end{array}$$ Verify the proposed rate law graphically (first determine what plot should yield a straight line), and calculate \(C_{\mathrm{A} 0}\) and \(k\) (c) Convert the formula with the calculated constants included to an expression for the molarity of A in the reaction mixture in terms of \(t\) (seconds). Calculate the molarity at \(t=265 \mathrm{s}\).

Step by step solution

01

Define Units of k

The units of \(k\) can be obtained from the formula \(C_A = C_{A_0} \exp (-k t)\). We solve for \(k\) to get \(k = - \dfrac{1}{t} \ln \left(\dfrac{C_A}{C_{A_0}}\right)\). So, the units of \(k\) would be \(\dfrac{lbmol}{ft^3 . min}\).

02

Establish Rate Law Graphically

To verify the rate law graphically, we plot \(C_A\) against \(t\). From the relation \(C_A = C_{A_0} \exp (-k t)\), the straight-line graph will have a slope of -\(k\). Thus, using this, \(C_{A_0}\) and \(k\) can be determined from the intercept and slope of the graph.

03

Convert to Molarity Expression and Calculation

The molarity (\(M_A\)) can be calculated using the relation \(M = \dfrac{n}{V}\). Given that \(C_A\) is in \(lbmol/ft^3\) and that \(1 ft^3 = 28.3168 L\), we can convert the expression of \(C_A\), which gives \(M_A(t) = C_{A_0} \exp [-k t (60 lbmol/ft^3)] / 28.3168\). Now, \(M_A\) at \(t=265 s\) can be calculated using \(M_A(265) = C_{A_0} \exp [-k (265/60) lbmol/ft^3 ] / 28.3168 \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Rate Law

In chemical reaction kinetics, a rate law is an equation that connects the rate of a chemical reaction to the concentration of its reactants. A common form of a rate law is

• For a reaction: \(A \rightarrow B\)
• The rate law could be: \(C_A = C_{A_0} \exp(-kt)\)

This implies that the concentration of A, \(C_A\), decreases exponentially over time, influenced by a rate constant \(k\).The rate constant \(k\) is key in defining how quickly the reaction progresses. Its units depend on the order of the reaction. For a first-order reaction like the one in this exercise, the units are time^−1 (e.g., \(min^{-1}\)) because the formula involves an exponential decay in concentration over time.Experimental data can verify the rate law. By plotting the natural log of concentration (\(\ln(C_A)\)) against time (\(t\)), a straight line with slope \(-k\) and intercept \(\ln(C_{A_0})\) is expected. This confirms an exponential decay model for the reaction.

Concentration Change

Concentration change describes how the amount of a reactant, \(C_A\), shifts over time during a reaction. This change can often be attributed to the reaction progressing towards completion, converting reactants into products.In the initial stage of a reaction where \(C_A = C_{A_0} \exp(-kt)\), the exponential term indicates that concentration decreases rapidly initially, and slows down as time progresses. This is characteristic of a first-order reaction which means the rate of reaction is directly proportional to the concentration of one reactant.Visualizing concentration change is best done through a graph of \(C_A\) versus \(t\). In this graph, the visual decline of \(C_A\) can provide insight into the speed of the reaction. Key points to note include:

• Starting concentration: \(C_{A_0}\)
• Rate of decrease: influenced by \(k\)
• Long-term behavior: a stable minimum concentration

These observations help to understand the dynamics of reaction rates and provide practical insights into controlling real-world chemical processes.

Exponential Decay

Exponential decay describes a situation where quantities decrease at a rate proportional to their current value. In chemical kinetics, this term is often used to model the decrease of reactants over time. For a reaction like \(A \rightarrow B\), exponential decay provides a clear picture of how concentration reduces.The formula \(C_A = C_{A_0} \exp(-kt)\) derives from this concept. Here, \(C_A\) decreases exponentially with respect to time. Exponential functions have the form \(e^{x}\), representing natural exponential changes. For chemical concentrations:

• The base of the exponent: \(e\) is simply Euler's number (roughly 2.7183)
• The exponent: \(-kt\) indicates the rate at which \(C_A\) is shrinking

Practical applications involve predicting how much reactant remains after a given time. This is crucial for planning reaction durations and understanding kinetics. Thus, knowing how to apply exponential decay formulas allows us to accurately assess and manipulate chemical reactions effectively.