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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 \(2.0\) and \(5 \times 10^{-4}\), respectively, for time expressed in seconds.

Short Answer

Expert verified
Answer: \(dX(t)/dt = 10^{-3}t\exp(-5 \times 10^{-4}t^2)\)

Step by step solution

01

Write down the Avrami equation and the given constants

The Avrami equation is: \(1 - X(t) = \exp(-kt^n)\) The given constants are: \(n = 2.0\) \(k = 5 \times 10^{-4}\)
02

Find the derivative of the Avrami equation with respect to time \(t\)

Differentiating both sides of the Avrami equation with respect to \(t\), we get: \(-\frac{dX(t)}{dt} = -\frac{d}{dt} \exp(-kt^n)\) Now, applying the chain rule to differentiate the exponential term: \(-\frac{dX(t)}{dt} = \exp(-kt^n) \cdot (-nk^{n-1}t^{n-1})\)
03

Plug the given constants into the derivative equation

Now, we have to substitute the given values of \(n\) and \(k\) into the equation: \(-\frac{dX(t)}{dt} = \exp(-5 \times 10^{-4}t^2) \cdot (-2(5 \times 10^{-4})^{2-1}t^{2-1})\) Simplify the equation: \(-\frac{dX(t)}{dt} = -10^{-3}t\exp(-5 \times 10^{-4}t^2)\)
04

Write the final expression for the reaction rate

The rate of the reaction that obeys Avrami kinetics with the given constants \(n\) and \(k\) is: \(\frac{dX(t)}{dt} = 10^{-3}t\exp(-5 \times 10^{-4}t^2)\) This equation represents the rate of the reaction, as a function of time \(t\).

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Most popular questions from this chapter

Figure \(10.40\) shows the continuous-cooling transformation diagram for a \(0.35 \mathrm{wt} \% \mathrm{C}\) ironcarbon alloy. Make a copy of this figure, and then sketch and label continuous-cooling curves to yield the following microstructures: (a) Fine pearlite and proeutectoid ferrite (b) Martensite (c) Martensite and proeutectoid ferrite (d) Coarse pearlite and proeutectoid ferrite (e) Martensite, fine pearlite, and proeutectoid ferrite

10 The kinetics of the austenite-to-pearlite transformation obeys the Avrami relationship. Using the fraction transformed-time data given here, determine the total time required for \(95 \%\) of the austenite to transform to pearlite. \begin{tabular}{cc} \hline Fraction Transformed & Time \((s)\) \\ \hline \(0.2\) & 280 \\ \hline \(0.6\) & 425 \\ \hline \end{tabular}

Name the microstructural products of 4340 alloy steel specimens that are first completely transformed to austenite, then cooled to room temperature at the following rates: (a) \(0.005^{\circ} \mathrm{C} / \mathrm{s}\) (b) \(0.05^{\circ} \mathrm{C} / \mathrm{s}\) (c) \(0.5^{\circ} \mathrm{C} / \mathrm{s}\) (d) \(5^{\circ} \mathrm{C} / \mathrm{s}\)

For some transformation having kinetics that obey the Avrami equation (Equation 10.17), the parameter \(n\) is known to have a value of \(1.5 .\) If the reaction is \(25 \%\) complete after \(125 \mathrm{~s}\), howlong (total time) will it take the transformation to go to \(90 \%\) completion?

Briefly explain why fine pearlite is harder and stronger than coarse pearlite, which in turn is harder and stronger than spheroidite.
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