Heat capacity of mixtures

Question

After reading this recent question I was interested in how to calculate the specific heat capacity of a mixture based on the specific heat capacities of its components.

According to this website the specific heat capacity of an ideal mixture is given by

$$C_{p(\mathrm{mixture})} = \sum\limits_iC_{p(i)}x_i$$

where $x_i$ is the mole fraction of each component.

1. Where does this come from and what are the assumptions involved in treating it as an ideal mixture?
2. What deviations from this are observed in non-ideal mixtures? Can you easily predict the deviations based on the mixture and/or predict the actual specific heat capacity?

Answer 1

Suppose you have $n_1$ mole of A and $n_2$ mole of B. If they are at temp. $T_1$ and you want to raise the temp. to $T_2$ then the amount of heat you need to add is
$$\Delta H~= n_1 \times c_{pA} \times \Delta T + n_2 \times c_{pB} \times \Delta T$$
$$\Delta H~= (n_1 \times c_{pA} + n_2 \times c_{pB}) \times \Delta T$$
$$\Delta H~= (n_1 +n_2)\times \frac{(n_1 \times c_{pA} + n_2 \times c_{pB})}{(n_1+n_2)} \times \Delta T$$
$$\Delta H~= n_{total} \times(y_1 \times c_{pA} + y_2 \times c_{pB}) \times \Delta T$$
$$\Delta H~= n_{total} \times c_{p,avg} \times \Delta T$$
So basically you are not losing any information in case of ideal situation where molecular interaction of two different species has no effect on each other's heat capacity. But from statistical mechanics we know, $$C_v=\frac { <E^2>-<E>^2}{k_B \times T^2}$$
Now if we solve the schrodinger equation for individual species, we must include an extra potential energy term to account the interaction from other species, which in turn will result in fluctuation of $C_v$ from it's previous value.

Slightly off topic: When I see the question first things that came to my mind is kopp's rule. It's used to find heat capacity of molecule using atom's heat capacity or to find alloy's heat capacity using the constituents heat capacity.