Solving 2-D Crystal Process Avrami Equation: Find k

[Stat313]

Homework Statement

A two dimensional crystal process is modeled by an Avrami equation phi(t)= 1-exp(-k*t^2), where k>0. Experimental observation suggests that the inflection point of phi occurs at t=2 hours. Find the dimension and numerical value of k.

Homework Equations

I know how to take the log-log of this equation in order to linearize it.

The Attempt at a Solution

I tried to have a system of two equations with 2 unknowns, but failed because I don't know the value of phi at any time t. Any help would be greatly appreciated.

 

[gneill]

Homework Statement

A two dimensional crystal process is modeled by an Avrami equation phi(t)= 1-exp(-k*t^2), where k>0. Experimental observation suggests that the inflection point of phi occurs at t=2 hours. Find the dimension and numerical value of k.

Homework Equations

I know how to take the log-log of this equation in order to linearize it.

The Attempt at a Solution

I tried to have a system of two equations with 2 unknowns, but failed because I don't know the value of phi at any time t. Any help would be greatly appreciated.

Inflection points are usually found using derivatives of the curve.

 

[Stat313]

Thanks for the reply,
I computed the second derivative with respect to t, I got
phi^2(t)= 2*k*exp(-k*t^2)-4*k^2*t^2*exp(-k*t^2)

Since we know that an inflection point occurs at t = 2, I solved for phi^2(2) = 0, which gives me k=0,1/8. so we choose k>0. Is my answer correct?

 

[gneill]

Thanks for the reply,
I computed the second derivative with respect to t, I got
phi^2(t)= 2*k*exp(-k*t^2)-4*k^2*t^2*exp(-k*t^2)

Since we know that an inflection point occurs at t = 2, I solved for phi^2(2) = 0, which gives me k=0,1/8. so we choose k>0. Is my answer correct?

Your method is fine. If the magnitude of k is 1/8, what units will k have?

 

[Stat313]

Can you elaborate more please?

 

[gneill]

The equation contains the exponential term
$$e^{-k\;t^2}$$
What units should be associated with k in order to make this term correct (mathematically, so that when evaluated numerically will yield a real number).

 

[Stat313]

Is't k just a constant?

 

[gneill]

Is't k just a constant?

Sure, but even constants have units if equations using them are to balance.

 

[Stat313]

When I compute phi(2), it gives me 0.39 which is a real number.

 

[Stat313]

Any hints?

 

[gneill]

Any hints?

Hint: The argument x of the exponential function $e^x$ must be a pure number without units (all units in the exponent must cancel).

 

[Stat313]

Is the dimension of k is (1/time(t)), in this case we have t in hours?

 

[gneill]

Is the dimension of k is (1/time(t)), in this case we have t in hours?

Test it. If k = 1/(8*hr), does the exponent yield a unitless number if you plug in t = 2 hr?

 

[Stat313]

No, it is not going to cancel, but if I have k 1/time in (half hours), it does cancel? By the way, I am not a physics major, so bare with me please.

 

[haruspex]

t has dimension time, so what dimension does t² have? So what dimension do you need k to have in order to produce a dimensionless kt²?

 

[gneill]

In the exponent you have the expression $-k t^2$. That means the time is squared, so if t is in hours, the units of $t^2$ are $hr^2$. To make the units of k cancel this, it should have units of $1/(hr^2)$, or $hr^{-2}$.

 

[Stat313]

t has dimension time, so what dimension does t² have? So what dimension do you need k to have in order to produce a dimensionless kt²?

t^2 has (time)^2 and k has to have (1/(time)^2) ?

 

[haruspex]

t^2 has (time)^2 and k has to have (1/(time)^2) ?

Yes. So the units would be?

 

[Stat313]

The units will cancel, (time)^2/(time)^2=1

Thanks for the help,

 

[haruspex]

The units will cancel, (time)^2/(time)^2=1

No, I meant the units of k.

 

[Stat313]

No, I meant the units of k.

Is it (1/t)^2?

 

[haruspex]

No, that's a dimension, like distance-squared, 1/time, mass... Units are things like sq metres, Herz, kgs...