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Chapter 11: Problem 47

The time taken in \(75 \%\) completion of a zero-order reaction is \(10 \mathrm{~h}\). In what time, he reaction will be \(90 \%\) completed? a) \(12.0 \mathrm{~h}\) (b) \(16.6 \mathrm{~h}\) c) \(10.0 \mathrm{~h}\) (d) \(20.0 \mathrm{~h}\)

Short Answer

Expert verified
The reaction will be 90% completed in 12.0 hours.

Step by step solution

01

Understand Zero-Order Reaction Kinetics

In a zero-order reaction, the rate of the reaction is constant and does not depend on the concentration of the reactants. The rate of reaction can be given by the equation: \[R = -k\], where \(R\) is the rate of reaction and \(k\) is the rate constant. The amount of reactant A remaining at any time \(t\) is given by \(A_t = A_0 - kt\), where \(A_0\) is the initial concentration and \(A_t\) is the concentration at time \(t\).
02

Calculate Time for 75% Completion

For 75% completion of the reaction, the remaining concentration of the reactant is 25% of the initial concentration. If we assume the initial concentration \(A_0\) to be 100 units for simplicity, then the remaining concentration \(A_t\) after 75% completion is 25 units. Using the relationship \(A_t = A_0 - kt\), we have \[25 = 100 - k(10)\]. From this equation, we can solve for the rate constant \(k\).
03

Calculate the Rate Constant (k)

Solving for \(k\) gives us \[k = \frac{100 - 25}{10} = \frac{75}{10} = 7.5 \frac{units}{h}\].
04

Calculate Time for 90% Completion

For 90% completion, 10% of the reactant remains. Using the same initial concentration of 100 units, \(A_t\) is 10 units. We use the equation \(A_t = A_0 - kt\) again, replacing \(A_t\) with 10 and \(k\) with the value we found: \[10 = 100 - 7.5t\]. Then we solve for \(t\) to find the time for 90% completion.
05

Solve for Time (t)

Rearranging the equation from the previous step gives us \[7.5t = 100 - 10\]. \[7.5t = 90\]. Dividing both sides by 7.5 gives us \[t = \frac{90}{7.5} = 12 \mathrm{~h}\].

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Key Concepts

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

Chemical Kinetics
Chemical kinetics is the study of the speed or rate at which chemical reactions occur. It's crucial to understanding how long it takes for a reaction to reach a certain point, such as the completion time for a reactant to be used up or for a product to be formed. In kinetics, we analyze the factors affecting reaction rates, investigating aspects such as concentration, temperature, surface area, and catalysts.

In the case of a zero-order reaction, the rate at which the chemicals react is independent of the concentration of the reactants. This counterintuitive scenario means that even as the amount of reactant decreases, the reaction continues at a steady pace until one of the reactants is depleted. It's the chemical kinetics principles that enable us to predict how long it will take for a reaction to progress to a certain point, supplying critical information for industrial processes, pharmaceuticals, and research.
Reaction Rate Constant
The reaction rate constant, symbolized as 'k', is a proportionality factor that plays a pivotal role in chemical kinetics. It connects the rate of the reaction to the concentrations of the reactants in a particular reaction order. The value of 'k' is determined by experimental measures and can be influenced by variables including temperature, the presence of catalysts, or the solvent in which the reaction is taking place.

For a zero-order reaction, the rate constant 'k' represents the rate of reaction itself because it is constant and independent of the concentration of reactants. The units for 'k' are typically concentration per unit time (such as Molarity per second, or units per hour as in our exercise). Knowing the rate constant allows us to calculate how long it will take for a reaction to reach a certain completion level, which in this context is related to the depletion of reactants over time.
Reaction Completion Time
Reaction completion time refers to the duration it takes for a reaction to reach a certain defined point of completion. This could be a specific percentage of reactant used or product formed. It's a direct application of chemical kinetics and relies heavily on the reaction rate constant 'k'.

By applying the formulas from chemical kinetics, one can determine the amount of time required for different extents of reaction completion. In our exercise, we used the zero-order reaction rate equation to calculate the time it would take for 90% completion after establishing the rate constant from the time taken for 75% completion. This exercise illustrates the direct, linear relationship between time and product formation in zero-order reactions - as time continues, the amount of reactant decreases linearly, simplifying the estimation of completion time.

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Most popular questions from this chapter

Rate of which type of elementary reaction increases with increase in temperature? (a) Thermal (b) Exothermic (c) Endothermic (d) All

The rate equation for an autocatalytic reaction \(\mathrm{A}+\mathrm{R} \stackrel{k}{\longrightarrow} \mathrm{R}+\mathrm{R}\) is \(r_{\mathrm{A}}=-\frac{\mathrm{d} C_{\mathrm{A}}}{\mathrm{d} t}=k C_{\mathrm{A}} C_{\mathrm{R}}\) The rate of disappearance of reactant \(\mathrm{A}\) is maximum when (a) \(C_{\mathrm{A}}=2 C_{\mathrm{R}}\) (b) \(C_{\mathrm{A}}=C_{\mathrm{R}}\) (c) \(C_{\mathrm{A}}=C_{\mathrm{R}} / 2\) (d) \(C_{\mathrm{A}}=\left(C_{\mathrm{R}}\right)^{1 / 2}\)

For the reaction: \(2 \mathrm{~N}_{2} \mathrm{O}_{5}(\mathrm{~g}) \rightarrow 4 \mathrm{NO}_{2}(\mathrm{~g})\) \(+\mathrm{O}_{2}(\mathrm{~g})\), the concentration of \(\mathrm{NO}_{2}\) increases by \(2.4 \times 10^{-2} \mathrm{M}\) in \(6 \mathrm{~s}\). What will be the average rate of appearance of \(\mathrm{NO}_{2}\) and the average rate of disappearance of \(\mathrm{N}_{2} \mathrm{O}_{5} ?\) (a) \(2 \times 10^{-3} \mathrm{Ms}^{-1}, 4 \times 10^{-3} \mathrm{Ms}^{-1}\) (b) \(2 \times 10^{-3} \mathrm{Ms}^{-1}, 1 \times 10^{-3} \mathrm{Ms}^{-1}\) (c) \(2 \times 10 \mathrm{Ms}^{-1}, 2 \times 10^{-3} \mathrm{Ms}^{-1}\) (d) \(4 \times 10^{-3} \mathrm{Ms}^{-1}, 2 \times 10^{-3} \mathrm{Ms}^{-1}\)

According to the collisions theory, the rate of reaction increases with temperature due to (a) increase in number of collisions between reactant molecules. (b) increase in speed of reacting molecules. (c) increase in number of molecules having sufficient energy for reaction. (d) decrease in activation energy of reaction.

In Lindemann theory of unimolecular reactions, it is shown that the apparent rate constant for such a reaction is \(k_{\text {app }}\) \(=\frac{k_{1} C}{1+\alpha C}\), where \(C\) is the concentration of the reactant, \(k_{1}\) and a are constants. The value of \(C\) for which \(k_{\text {app }}\) has \(90 \%\) of its limiting value at \(C\) tending to infinitely large is \(\left(\alpha=9 \times 10^{5}\right)\) (a) \(10^{-6}\) mole/litre (b) \(10^{-4}\) mole/litre (c) \(10^{-5}\) mole/litre (d) \(5 \times 10^{-5}\) mole/litre
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