The excess Gibbs energy of mixing for a regular solution can be derived systematically by defining the interaction energy for particle groups composed of closest neighbors at equivalent site. The internal energy of a system is regarded as the sum of the interaction energy for particle groups and the excess function is derived from the sum of the interaction energy change caused by the formations of particle groups. In the present paper, we have derived the excess Gibbs energy of mixing for ternary and quaternary regular solutions with a closest packing. For a ternary regular solution, the molar excess Gibbs energy of mixing is given as follows, G
ex=XAXB(XAWAAB+XBWABB)+XBXC (XBWBBC+XCWBCC) +XCXA(XCWCCA+XAWCAA)+2XAXBXCWABC, where
Xi is a mole fraction of the component
i and
Wijk is an interaction parameter of [ijk] triplet. The following expressions for
n-component systems are derived by the similar method by supposing interaction energy among
r particles. Some previously proposed models can be derived from our model.
n=2,
r=2: G
ex=XAXBWAB (
symmetric binary regular solution model)
n=2,
r=3: G
ex=XAXB(XAWAAB+XBWABB)(
asymmetvie binary regular solution model)
n=3,
r=2: G
ex=XAXBWAB+XBXCWBC+XCXAWCA (
symmetric ternary regular solution model)
n=3,
r=3: G
ex=XAXB(XAWAAB+XBWABB)+XBXC(XBWBBC+XCWBCC) +XCXA(XCWCCA+XAWCAA)+2XAXBXCWABC (
new ternary model: stated above)
n=4,
r=4: G
ex=XA
3XBWAAAB+XB
3XAWBBBA+XA
3XCWAAAC+XC
3XAWCCCA +XA
3XDWAAAD+XD
3XAWDDDA+XB
3XCWBBBC+XC
3XBWCCCB +XB
3XDWBBBD+XD
3XBWDDDB+XC
3XDWCCCD+XD
3XCWDDDC +
3/2XA
2XB
2WAABB+
3/2XA
2XC
2WAACC+
3/2XA
2XD
2WAADD +
3/2XB
2XC
2WBBCC+
3/2XB
2XD
2WBBDD+
3/2XC
2XD
2WCCDD +3XA
2XBXCWAABC+3XA
2XBXDWAABD+3XA
2XCXDWAACD +3XB
2XAXCWBBAC+3XB
2XAXDWBBAD+3XB
2XCXDWBBCD +3XC
2XAXBWCCAB+3XC
2XAXDWCCAD+3XC
2XBXDWCCBD +3XD
2XAXB⊗WDDAB+3XD
2XAXCWDDAC+3XD
2XBXCWDDBC +6XAXBXCXDWABCD.
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