1.90: Einstein's Heat Capacity Equation Fit to Experimental Data - Another Algorithm
- Page ID
- 158505
\[
n :=30 \qquad \mathrm{i} :=1 \ldots \mathrm{n}
\nonumber \]
\(\mathrm{T}_{\mathrm{i}} :=\) | \(\mathrm{C}_{\mathrm{i}} :=\) |
1 | 0.000818 |
3 | 0.0065 |
5 | 0.0243 |
8 | 0.0927 |
10 | 0.183 |
15 | 0.670 |
20 | 1.647 |
25 | 3.066 |
30 | 4.774 |
35 | 6.612 |
40 | 8.419 |
45 | 10.11 |
50 | 11.66 |
55 | 13.04 |
60 | 14.27 |
65 | 15.35 |
70 | 16.30 |
80 | 17.87 |
90 | 19.11 |
100 | 20.10 |
120 | 21.54 |
140 | 22.52 |
160 | 23.22 |
180 | 23.75 |
200 | 24.16 |
220 | 24.49 |
240 | 24.76 |
260 | 24.99 |
280 | 25.19 |
300 | 25.37 |
The heat capacity data were taken from the Handbook of Physics and Chemistry - 72nd Edition, page 5-71. The data are presented in units of Joules/mole/K.
\[
\mathrm{R} :=8.3145
\nonumber \]
Define Einstein function for heat capacity and first derivative with respect \(\Theta\):
\[
F(T, \Theta) :=\left[\begin{array}{c}{3 \cdot R \cdot\left(\frac{\Theta}{T}\right)^{2} \cdot \frac{\exp \left(\frac{\Theta}{T}\right)}{\left(\exp \left(\frac{\Theta}{T}\right)-1\right)^{2}}} \\ {\frac{d}{d \Theta}\left[3 \cdot R \cdot\left(\frac{\Theta}{T}\right)^{2} \cdot \frac{\exp \left(\frac{\Theta}{T}\right)}{\left(\exp \left(\frac{\Theta}{T}\right)-1\right)^{2}}\right]}\end{array}\right]
\nonumber \]
Call genfit to do nonlinear regression analysis.
\[
P :=\text { genfit }(T, C, 200, F) \qquad P=154.707
\nonumber \]
Plot data and fit:
\[
t :=1 \ldots 300
\nonumber \]