I am trying to calculate the parameters of the Vogel−Fulcher−Tamman Equation. $T$ is temperature and $a,b,c$ are constants to be found.
$$\eta = a\exp(\frac{b}{T-c})$$
The experimental values are below
| Temp /k | Viscosity $\eta$ |
|---|---|
| 293 | 313.9 |
| 313 | 75.6 |
| 373 | 7.6 |
I have tried the following: $$\ln \eta = \ln a + \frac{b}{T-c}$$
If it was $\ln \eta = \ln a + \frac{b}{T}$, I could plot $\ln \eta$ against $\frac{1}{T}$ and the gradient would give b and the y-intercept would give me $\ln a$.
I don't know how to account for the $T-c$ in the denominator?
I don't think there's a closed form solution.
You have three equations in three unknowns $x$, $y$ and $z$ of the form $$ x - r_i = \frac{y}{s_i - z} $$ for known values corresponding to $i = 1,2,3$.
If you subtract the first from the second and the third you have two equations
$$ r_1 - r_2 = y\left(\frac{1}{s_2-z} - \frac{1}{s_1-z} \right) $$ $$ r_1 - r_3 = y\left(\frac{1}{s_3-z} - \frac{1}{s_1-z} \right) . $$ Then the ratio of those two equations is a single ugly nonlinear equation in $z$.
Solve it numerically and work backwards to get $y$ and then $x$.