Thermodynamic properties

Volume

If we differentiate this endure blueprint with account to P at T connected we get:

\left(\frac{\partial g(T,P)}{\partial P}\right)_{T}=RT\left(\frac{\partial \ln f}{\partial P}\right)_{T}

but we apperceive from the Gibbs abeyant blueprint that:

\left(\frac{\partial g(T,P)}{\partial P}\right)_{T}=v

These endure two equations put calm give:

\left(\frac{\partial \ln f}{\partial P}\right)_{T}=\frac{v}{RT}

Since all this, done as a authentic actuality is accurate in a mix just abacus the subscript i to all the accelerated variables and alteration v to \bar{v_i}, continuing for Partial molar volume.

\left(\frac{\partial \ln f_i}{\partial P}\right)_{T,x_i}=\frac{\bar{v_i}}{RT}

Applying the aboriginal blueprint of this area to this endure blueprint we get

v_i^*=\bar{v_i}

which agency that in an ideal mix the aggregate is the accession of the volumes of its components.

edit Enthalpy and calefaction capacity

Proceeding in a agnate way but acquired with account of T we get to a agnate aftereffect with enthalpies

\frac{g(T,P)-g^\mathrm{gas}(T,p^u)}{RT}=\ln\frac{f}{p^u}

derivative with account to T and canonizing that \left( \frac{\partial \frac{g}{T}}{\partial T}\right)_P=-\frac{h}{T^2} we get:

-\frac{\bar{h_i}-h_i^\mathrm{gas}}{R}=-\frac{h_i^*-h_i^\mathrm{gas}}{R}

which in about-face is \bar{h_i}=h_i^*.

Meaning that the enthalpy of the mix is according to the sum of its components.

Since \bar{u_i}=\bar{h_i}-p\bar{v_i} and u_i^*=h_i^*-pv_i^*:

u_i^*=\bar{u_i}

It is aswell calmly absolute that

C_{pi}^*=\bar{C_{pi}}

edit Anarchy of mixing

Finally since

\bar{g_i}=\mu _i=g_i^\mathrm{gas}+RT\ln \frac{f_i}{p^u}=g_i^\mathrm{gas}+RT\ln \frac{f_i^*}{p^u}+RT\ln x_i=\mu _i^*+ RT\ln x_i

Which agency that

Δgi,mix = RTln xi

and since

G = ∑ xigi

i

then

ΔGmix = RT ∑ xiln xi

i

At endure we can account the anarchy of bond back g_i^*=h_i^*-Ts_i^* and \bar{g_i}=\bar{h_i}-T\bar{s_i}

Δsi,mix = − R ∑ ln xi

i

ΔSmix = − R ∑ xiln xi