## CpFit

CpFit program implements thermodynamic model of heat capacity and other thermodynamic functions based on usage of Einstein-Plank functions sum and empirical parameters. The used approach has been developed by Voronin et al. [1].

All thermodynamic functions (e.g. heat capacity, entropy and enthalpy) are represented as sums of Einstein-Plank terms:

$C_p(T) = \sum_{i=1}^m \alpha_i C_\textrm{E}\left(\frac{\theta_i}{T}\right)$$\frac{C_\textrm{E}(x)}{R} = \frac{3x^2e^x}{\left(e^x - 1\right)^2}$

$S(T) - S(0) = \sum_{i=1}^m \alpha_i S_\textrm{E}\left(\frac{\theta_i}{T}\right)$$\frac{S_\textrm{E}(x)}{R} = 3\left[\frac{x}{e^{x} - 1} - \ln\left(1 - e^x\right)\right]$

$H(T) - H(0) = \sum_{i=1}^m \alpha_i \left[U_\textrm{E}\left(\frac{\theta_i}{T}\right) - U_0 \right]$$\frac{U_\textrm{E}(x) - U_0}{RT} = \frac{3x}{e^x - 1}$

where $\alpha_i$ and $\theta_i$ are adjustable (usually by means of the least squares method) model parameters. They can be estimated from experimental data using least squares method. A universal gas constant value $R=8.3144598~\mathrm{J\cdot mol^{-1} K^{-1}}$ from CODATA 2014 is used in the CpFit program. In the eqs $\alpha_i$ values are dimensionless and $\theta_i$ values are expressed in K.

The thermodynamic model of heat capacity based on Einstein-Plank functions allows to approximate $C_p(T)$, $S(T)-S(0)$ and $H(T)-H(0)$ in a wide temperature range using a uniﬁed set of parameters. Also unlike polynomial models it can give a physically correct extrapolation to a wider temperature range.

Implementation of this model in CpFit hides all analytical expressions and technical details and allow to use it as an ordinary statistical package for a nonlinear regression.

CpFit main window

CpFit results window

Approximation of heat capacity for natrolite

[1] Gennady F. Voronin, Ilya B. Kutsenok Universal Method for Approximating the Standard Thermodynamic Functions of Solids // J. Chem. Eng. Data, 2013, 58, 2083−2094