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I was wondering:

How is the specific heat of an alloy obtained?

My first attempt is to assume a thermal equilibrium between a mechanical solution and an homogeneous solution of two metals, A and B.

\begin{align*} Q_{A+B}&=Q_A+Q_B\\ n_{A+B}C_{A+B}(T_{eq}-T_{A+B})&=n_{A}C_{A}(T_{eq}-T_A)+n_{B}C_{B}(T_{eq}-T_B)\\ C_{A+B}(T_{eq}-T_{A+B})&=\frac{n_{A}}{n_{A+B}}C_{A}(T_{eq}-T_A)+\frac{n_{B}}{n_{A+B}}C_{B}(T_{eq}-T_B)\\ C_{A+B}(T_{eq}-T_{A+B})&=X_AC_{A}(T_{eq}-T_A)+X_BC_{B}(T_{eq}-T_B)\\ C_{A+B}&=\frac{X_AC_{A}(T_{eq}-T_A)+X_BC_{B}(T_{eq}-T_B)}{(T_{eq}-T_{A+B})} \end{align*} Where Q=heat, C=specific heat, n=moles, X=molar fraction and T=temperature.

Is it correct?

Is there another way of doing this?

I have previously searched and this are some relevant things on Chem.SE what I have found:

I did not find a solution or an approximation in any of those links.

My aim is to prepare a class and teach my students how to determine the specific heat of any alloy, if possible, without going to a lab. (There is no basic infrastructure for special labs).

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The enthalpies and calorific capacities of certain mixtures are tabulated in standard references. If you do not have such data, you can use the following approximation:

Rule 1: For a mixture of gases or liquids, calculate the total change in enthalpy as the sum of its changes in the pure components of the mixture. In fact, the enthalpy changes associated with component mixing are neglected, which is an excellent approximation for gas mixtures and similar liquid mixtures such as pentane and hexane, but is bad for different liquids, such as nitric acid and Water.

Rule 2: For highly diluted solutions of solids or gases in liquids, neglect the change in enthalpy of the solute. This approach is better, as the solution is more diluted.

The calculation of the enthalpy changes for heating or cooling a known composition mixture can often be simplified by estimating the calorific capacity of the mixture as follows: \begin{equation} C_{p,m}(T) = \sum_i y_i\cdot C_{p,i}(T) \end{equation}

Where:

$C_{p,m}(T)$ Calorific capacity of the mixture

$y_i$ Mass or mole fraction of the i-th component

$C_{p,i}(T)$ Calorific capacity of the i-th component

Reference:

Richard Mark Felder and Ronald W. Rousseau. Elementary Principles of Chemical Processes. John Wiley & Sons Inc.; 3rd Revised edition. 702 pages.

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