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Chapter 10: Problem 7

Compute the rate of some reaction that obeys Avrami kinetics, assuming that the constants \(n\) and \(k\) have values of \(3.0\) and \(7 \times 10^{-3}\), respectively, for time expressed in seconds.

Short Answer

Expert verified
Question: Calculate the rate of reaction for a process following Avrami kinetics with constants n = 3.0 and k = 7 × 10^-3. Answer: The rate of reaction for the given Avrami kinetics can be determined using the equation dY(t)/dt = 3(7 × 10^{-3})^3 t^2 e^{-(7 × 10^{-3}t)^3}.

Step by step solution

01

Write down the Avrami equation

The Avrami equation is given by the following expression: \(Y(t) = 1 - e^{-(kt)^n}\) Where \(Y(t)\) is the extent of the reaction at time t, \(k\) is the rate constant, \(t\) is the time, and \(n\) is the Avrami exponent.
02

Insert given values

We are given the values for the constants \(n = 3.0\) and \(k = 7 \times 10^{-3}\). Substitute these values into the Avrami equation: \(Y(t) = 1 - e^{-(7 \times 10^{-3}t)^3}\)
03

Calculate the rate of reaction

To compute the rate of the reaction, we need to derive the equation with respect to time. Calculate the derivative of the Avrami equation with respect to time \(t\): \(\frac{dY(t)}{dt} = \frac{d}{dt}\left(1 - e^{-(7 \times 10^{-3}t)^3}\right)\) Now, apply the chain rule to calculate the derivative: \(\frac{dY(t)}{dt} = -e^{-(7 \times 10^{-3}t)^3}\cdot(-3\cdot(7 \times 10^{-3})^3\cdot t^2)\) Simplify the equation: \(\frac{dY(t)}{dt} = 3(7 \times 10^{-3})^3 t^2 e^{-(7 \times 10^{-3}t)^3}\) This is the equation for the rate of reaction under Avrami kinetics for the given values of constants \(n\) and \(k\).

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Most popular questions from this chapter

For a eutectoid steel, describe isothermal heat treatments that would be required to yield specimens having the following Rockwell hardnesses: (a) \(93 \mathrm{HRB}\) (b) \(40 \mathrm{HRC}\) (c) \(27 \mathrm{HRC}\)

Name the microstructural products of eutectoid iron-carbon alloy \((0.76 \mathrm{wt} \% \mathrm{C})\) specimens that are first completely transformed to austenite, then cooled to room temperature at the following rates: (a) \(200^{\circ} \mathrm{C} / \mathrm{s}\), (b) \(100^{\circ} \mathrm{C} / \mathrm{s}\), and (c) \(20^{\circ} \mathrm{C} / \mathrm{s}\).

(a) Briefly describe the phenomena of superheating and supercooling. (b) Why do these phenomena occur?

Briefly describe the simplest continuous cooling heat treatment procedure that would be used in converting a 4340 steel from one microstructure to another. (a) (Martensite + bainite) to (ferrite + pearlite) (b) (Martensite + bainite) to spheroidite (c) (Martensite + bainite) to (martensite + bainite \(+\) ferrite)

The fraction recrystallized-time data for the recrystallization at \(600^{\circ} \mathrm{C}\) of a previously deformed steel are tabulated here. Assuming that the kinetics of this process obey the Avrami relationship, determine the fraction recrystallized after a total time of \(22.8 \mathrm{~min} .\) $$ \begin{array}{cc} \hline \text { Fraction Recrystallized } & \text { Time (min) } \\ \hline 0.20 & 13.1 \\ 0.70 & 29.1 \\ \hline \end{array} $$
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