Question

52-56. In this section, several models are presented and the solution of the associated differential equation is given. Later in the chapter, we present methods for solving these differential equations.

(Use of Tech) Chemical rate equations The reaction of certain chemical compounds can be modeled using a differential equation of the form y'(t) = -kyⁿ(t), where y(t) is the concentration of the compound, for t ≥ 0, k > 0 is a constant that determines the speed of the reaction, and n is a positive integer called the order of the reaction. Assume the initial concentration of the compound is y(0) = y₀ > 0.

c. Let y₀ = 1 and k = 0.1. Graph the first-order and second-order solutions found in parts (a) and (b). Compare the two reactions.

Accepted Answer

Recall the general differential equation for the chemical reaction: $$y'(t) = -k y$$^{n}$$(t)$$, where $$k \u003e 0$$ and $$n is $$the order of the reaction. For the first-order reaction ($$n=1$$), the differential equation becomes $$y'(t) = -k y(t). $$This is a separable differential equation. Solve the first-order equation by separating variables: write $$\frac{dy}{dt} = -k y$$, then rearrange to $$\frac{dy}{y} = -k dt. $$Integrate both sides to find the solution involving an exponential decay. For the second-order reaction ($$n=2$$), the differential equation is $$y'(t) = -k y$$^{2}$$(t). $$Again, separate variables: $$\frac{dy}{dt} = -k y$$^{2}$$ leads to $$\frac{dy}{$$y^{2}$$} = -k dt$$. Integrate both sides to find the solution, which will be a rational function in t$. With initial condition y(0$$) = $$y_0 = 1$ and k = 0.1$, write explicit expressions for both solutions (first-order and second-order). Then, plot both functions on the same graph for t$$ \geq 0$$ to compare how the concentration decreases over time for each reaction order.

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주요 개념

Differential Equations in Chemical Kinetics

Order of Reaction and Its Effect

Solving and Graphing Differential Equations