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1.90: Einstein's Heat Capacity Equation Fit to Experimental Data - Another Algorithm

  • Page ID
    158505
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    \[
    n :=30 \qquad \mathrm{i} :=1 \ldots \mathrm{n}
    \nonumber \]

    \(\mathrm{T}_{\mathrm{i}} :=\) \(\mathrm{C}_{\mathrm{i}} :=\)
    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:

    \[
    t :=1 \ldots 300
    \nonumber \]

    clipboard_e578469f86fada524ca6c30af38ce4094.png


    This page titled 1.90: Einstein's Heat Capacity Equation Fit to Experimental Data - Another Algorithm is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by Frank Rioux via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.