# 1.91: Fitting Debye's Heat Capacity Equation to Experimental Data for Silver

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$\mathrm{n} :=30 \qquad \mathrm{i} :=1 \ldots \mathrm{n} \nonumber$

 $$\mathrm{T}_{\mathrm{i}} :=$$ $$\mathrm{C}_{\mathrm{i}} :=$$ 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.

Gas law constant:

$\mathrm{R} :=8.31451 \nonumber$

Define Debye function for heat capacity:

$\mathrm{F}(\mathrm{T}, \Theta) :=9 \cdot \mathrm{R} \cdot\left(\frac{\mathrm{T}}{\Theta}\right)^{3} \cdot \int_{0}^{\frac{\Theta}{\mathrm{T}}} \frac{x^{4} \cdot \exp (\mathrm{x})}{(\exp (\mathrm{x})-1)^{2}} \mathrm{dx} \quad \text{where} x = \frac{hv}{kT} \nonumber$

Form the sum of the squares of the deviations:

$\operatorname{SSD}(\Theta) :=\sum_{\mathrm{i}}\left(\mathrm{C}_{\mathrm{i}}-\mathrm{F}\left(\mathrm{T}_{\mathrm{i}}, \Theta\right)\right)^{2} \nonumber$

Minimize the sum of the squares of the deviations:

$\Theta :=200 \nonumber$

Given

$\operatorname{SSD}(\Theta)=0 \qquad \Theta :=\operatorname{Minerr}(\Theta) \nonumber$

Debye Temperature for best fit:

$\Theta=210.986 \nonumber$

Mean squared error:

$\frac{\operatorname{SSD}(\Theta)}{(n-2)}=0.16 \nonumber$

Plot data and fit:

$t :=1 \ldots 300 \nonumber$

This page titled 1.91: Fitting Debye's Heat Capacity Equation to Experimental Data for Silver 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.