Heat capacities and standard entropies and enthalpies of some compounds essential for steelmaking and refractory design approximated by Debye-Einstein integrals

Ogris, Daniel Marian; Gamsjäger, Ernst

doi:10.1016/j.calphad.2021.102345

Cited by 2 publications

(3 citation statements)

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“…In our study, we employ the Debye-Einstein integral fit (Equation (1)) proposed by Wu et al [10] and extend its application to the entire measurement range (see also Gamsjäger and Wiessner [12] and Ogris and Gamsjäger [29]):…”

Section: Debye-einstein Integralmentioning

confidence: 99%

“…In our study, we employ the Debye–Einstein integral fit (Equation ( 1 )) proposed by Wu et al [ 10 ] and extend its application to the entire measurement range (see also Gamsjäger and Wiessner [ 12 ] and Ogris and Gamsjäger [ 29 ]):

where the Debye integral

is given by:

and the two Einstein terms

and

are given by:

These equations involve the following fit parameters: the Debye temperature

, the Einstein temperatures

with

, and the prefactors m ,

and

. According to theory, the sum of the prefactors are equal to the number of atoms in the formula unit as can be found in [ 5 ].…”

Section: Literature Data and Theorymentioning

confidence: 99%

“…It is worth noting that the Debye–Einstein integral approach allows for extrapolation to absolute zero in the case that ultra-low temperature data are missing; e.g., data are compiled for

only (see, e.g., [ 9 , 29 ]). The prefactors m ,

and

describe the weight of the Debye integral and the Einstein terms, respectively, where the sum is not a priori fixed in the fitting algorithm, but appears to be close to the number of atoms in the formula unit of the investigated compound as it should be from a theoretical point of view (see also [ 12 ]).…”

Section: Literature Data and Theorymentioning

confidence: 99%

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Extended Regression Analysis for Debye–Einstein Models Describing Low Temperature Heat Capacity Data of Solids

Gamsjäger,

Wiessner

2024

Entropy

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Heat capacity data of many crystalline solids can be described in a physically sound manner by Debye–Einstein integrals in the temperature range from 0K to 300K. The parameters of the Debye–Einstein approach are either obtained by a Markov chain Monte Carlo (MCMC) global optimization method or by a Levenberg–Marquardt (LM) local optimization routine. In the case of the MCMC approach the model parameters and the coefficients of a function describing the residuals of the measurement points are simultaneously optimized. Thereby, the Bayesian credible interval for the heat capacity function is obtained. Although both regression tools (LM and MCMC) are completely different approaches, not only the values of the Debye–Einstein parameters, but also their standard errors appear to be similar. The calculated model parameters and their associated standard errors are then used to derive the enthalpy, entropy and Gibbs energy as functions of temperature. By direct insertion of the MCMC parameters of all 4·105 computer runs the distributions of the integral quantities enthalpy, entropy and Gibbs energy are determined.

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Section: Debye-einstein Integralmentioning

confidence: 99%

where the Debye integral

is given by:

and the two Einstein terms

and

are given by:

These equations involve the following fit parameters: the Debye temperature

, the Einstein temperatures

with

, and the prefactors m ,

and

. According to theory, the sum of the prefactors are equal to the number of atoms in the formula unit as can be found in [ 5 ].…”

Section: Literature Data and Theorymentioning

confidence: 99%

“…It is worth noting that the Debye–Einstein integral approach allows for extrapolation to absolute zero in the case that ultra-low temperature data are missing; e.g., data are compiled for

only (see, e.g., [ 9 , 29 ]). The prefactors m ,

and

Section: Literature Data and Theorymentioning

confidence: 99%

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