Aluminum oxide equation

Goals: Heat capacity data for Al2O3 over the temperature range 15 K to 300 K is analyzed to obtain the standard molar entropy of this substance.

Prerequisites: A basic knowledge of entropy and related thermodynamic principles.

Resources you will need: This exercise should be carried out within a quantitative analysis software environment that is capable of fitting a polynomial function of arbitrary order to an x-y data set. You will also be graphing the data along with the fitted function.

Standard molar entropy is one of the most useful types of thermodynamic data for a substance. But how does one measure this property? Unfortunately, there is no instrument that directly measures entropy. Instead, values are determined indirectly through the measurement of constant pressure heat capacities (Cp).
To understand how this works, we start with the Clausius Inequality
where the equality in equation (1) only applies to a reversible process. The slow heating of a substance at constant pressure so that the sample is always in quasi-static thermal equilibrium with the surroundings is approximately reversible. In such a case, equation (1) becomes (2)
where we have rewritten the differential heat transfer in terms of the constant pressure heat capacity times the differential temperature change.
Imagine slowly heating a pure solid substance at constant pressure from absolute zero up to some arbitrary temperature, T. Integration of equation (2) over the full temperature range will yield the absolute entropy of the substance relative to absolute zero, (3)
The 3rd Law of Thermodynamics defines entropy to be zero for any pure crystalline solid at absolute zero, so the term can be dropped from equation (3).
Whenever a substance in thermal equilibrium with its surroundings undergoes a phase transition at constant pressure, the corresponding entropy change given by (4)
If any phase transitions occur as we are heating a substance from absolute zero, each will contribute to the overall absolute entropy of the substance. A refinement of equation (3) that accounts for melting and vaporization is given by
Equation (5) provides a means of determining the standard molar entropy of most substances. The terms corresponding to heating the solid, liquid, and gas phases, described by the 1st, 3rd, and 5th terms on the right-hand-side of equation (5), respectively, can be evaluated experimentally by measuring the heat capacity of the three phases over the appropriate temperature ranges, plotting these data as Cp/T verses T, and measuring the area under the curve.

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