Entropy of mixing

src: nanohub.org

In thermodynamics, entropy mixing is the increase in total entropy when some of the initial separate systems of different compositions, each in a thermodynamic state of internal balance, are mixed without chemical reactions by the impermeable thermodynamic deletion of the partition between them, followed by time for the formation of a new thermodynamic state of internal balance in a new unpartitioned closed system.

In general, mixing may be restricted to occur under various prescribed conditions. Under generally defined conditions, each of the starting materials at the same temperature and pressure, and the new system can change its volume, while maintained at the same constant temperature, pressure, and mass of chemical components. The volume available for each material to be explored increases, starting from the initially separated compartment, to the total general end volume. The latter volume should not be an initially separate volume amount, so work can be performed on or with a new closed system during the mixing process, as well as heat transferred to or from the vicinity, due to constant pressure and temperature maintenance.

The internal energy of the new closed system is equal to the amount of internal energy of the initially separate system. The reference value for internal energy should be determined in a restricted way to make it so, maintaining also that the internal energy is each proportional to the mass of the system.

For the conclusions in this article, the term 'ideal material' is used to refer to an ideal gas (mix) or ideal solution.

In the special case of ideal material mixing, the final general volume is actually the number of initial separated compartment volumes. There is no heat transfer and no work is done. The mixing entropy is fully accounted for by the diffusive expansion of each material into an inaccessible final volume.

In the general case of non-ideal mixing of materials, however, the total of the general final volume may differ from the number of separate initial volumes, and there may be movement of work or heat, to or from the surroundings; there may also be a mixed entropy departure from the corresponding ideal case. The departure is the main reason for interest in mixing entropy. Their energy and entropy variables and temperature dependence provide valuable information about the properties of the material.

At the molecular level, mixing entropy is interesting because it is a macroscopic variable that provides information about the properties of constitutive molecules. In an ideal material, the intermolecular forces are the same between each pair of molecular types, so the molecule does not feel the difference between the other molecules of its own kind and other types. In non-ideal materials, there may be differences in intermolecular forces or certain molecular effects between different species, although chemically unreacted. The mixing entropy provides information about the constitutive differences of intermolecular forces or specific molecular effects in the material.

The concept of random statistics is used for statistical mechanical explanations of mixing entropy. The mixing of ideal materials is considered random at the molecular level, and, thus, mixing of non-ideal materials may not be random.

Video Entropy of mixing

Mixing the ideal species at constant temperature and pressure

In an ideal species, the intermolecular force is the same between each pair of molecular types, so that the molecule "feels" there is no difference between itself and its molecular neighbor. This is a reference case for checking the appropriate mixing of non-ideal species.

For example, two ideal gases, at the same temperature and pressure, are initially separated by division of the partition.

Setelah penghapusan pembagian partisi, mereka memperluas ke volume umum akhir (jumlah dari dua volume awal), dan entropi pencampuran ${\ displaystyle \ Delta _ {mix} S \,}  ÂÂ$ diberikan oleh

${\ displaystyle \ Delta _ {mix} S = -nR (x_ {1} \ ln x_ {1} x_ {2} \ ln x_ {2}) \, }  ÂÂ$ .

In this case, the entropy increase is entirely due to the irreversible process of the expansion of the two gases, and does not involve heat or work flow between the system and its surroundings.

Gibbs free energy mixing

Energi bebas Gibbs mengubah $            {\displaystyle         {\mathrm{?}}_{            m            \mathrm{saya}            x          }        G        =        {\mathrm{?}}_{            m            \mathrm{saya}            x          }        H        -        T        {\mathrm{?}}_{            m            \mathrm{saya}            x          }        S              }        \{\backslash \; displaystyle\; \backslash \; Delta\; \_\; \{mix\}\; G\; =\; \backslash \; Delta\; \_\; \{mix\}\; H-T\; \backslash \; Delta\; \_\; \{mix\}\; S\; \backslash ,\}\;  ÂÂ$ menentukan apakah mencampur pada suhu konstan (absolut) ${\ displaystyle \ T}  ÂÂ$ dan tekanan ${\ displaystyle \ p}  ÂÂ$ adalah proses spontan. Kuantitas ini menggabungkan dua efek fisik - entalpi pencampuran, yang merupakan ukuran perubahan energi, dan entropi pencampuran yang dipertimbangkan di sini.

Untuk campuran gas ideal atau solusi ideal, tidak ada entalpi pencampuran ( ${\ displaystyle \ Delta _ {mix} H \,}  ÂÂ$ ), sehingga energi bebas Gibbs pencampuran diberikan oleh istilah entropi saja:

${\ displaystyle \ Delta _ {mix} G = -T \ Delta _ {mix} S \,}  ÂÂ$

For the ideal solution, Gibbs free energy mixing is always negative, which means that ideal mixing solutions are always spontaneous. The lowest value is when the mole fraction is 0.5 to mix the two components, or 1/n for the n component mix.

Maps Entropy of mixing

Solution and temperature dependency miscibility

The ideal and regular solution

The above equation for the ideal gas mixing entropy also applies to a particular solution liquid (or solid) - formed by completely random mixing so that the component moves independently in total volume. Mixing such a random solution occurs if the interaction energy between unequal molecules is similar to the average interaction energy among similar molecules. The value of entropy is exactly the same as random mixing for the ideal solution and for the regular solution, and roughly so many real solutions.

Untuk campuran biner entropi pencampuran acak dapat dianggap sebagai fungsi dari fraksi mol satu komponen.

${\ displaystyle \ Delta _ {mix} S = -nR (x_ {1} \ ln x_ {1} x_ {2} \ ln x_ {2}) = - nR [x \ ln x (1-x) \ ln (1-x)] \,}  ÂÂ$

Untuk semua kemungkinan campuran, ${\ displaystyle 0 & lt; x & lt; 1}  ÂÂ$ , sehingga ${\ displaystyle \ ln}  ÂÂ$ ${\ displaystyle x}  ÂÂ$ dan ${\ displaystyle \ ln (1-x)}  ÂÂ$ keduanya negatif dan entropi pencampuran ${\ displaystyle \ Delta _ {mix} S \,}  ÂÂ$ positif dan mendukung pencampuran komponen murni.

Juga kelengkungan ${\ displaystyle \ Delta _ {mix} S \,}  ÂÂ$ sebagai fungsi ${\ displaystyle x}  ÂÂ$ diberikan oleh turunan kedua ${\ displaystyle \ left ({\ frac {\ partial ^ {2} \ Delta _ {mix} S} {\ partial x ^ {2}}} \ right) _ {T, P} = - nR \ left ({\ frac {1} {x}} {\ frac {1} {1-x}} \ right)}  ÂÂ$

Kelengkungan ini negatif untuk semua campuran yang mungkin ${\ displaystyle (0 & lt; x & lt; 1)}  ÂÂ$ , sehingga mencampur dua solusi untuk membentuk solusi komposisi menengah juga meningkatkan entropi sistem. Pencampuran acak karena itu selalu mendukung miscibility dan menentang pemisahan fasa.

For an ideal solution, the enthalpy mixing is zero so the components can be dissolved in all proportions. For regular solutions, positive enthalpy mixing can cause imperfect miscibility (phase separation for some compositions) at temperatures below the critical critical temperature (UCST) temperature. This is the minimum temperature where ${\ displaystyle -T \ Delta _ {mix} S \,}$ The term in energy mixing Gibbs is enough to produce miscibility in all proportions.

Systems with lower critical temperature

Non-random mixing with lower mixing entropy can occur when interaction attracts between unequal molecules is significantly stronger (or weaker) than the average interaction between such molecules. For some of these systems it may cause a lower critical solution temperature (LCST) or a lower temperature limiting for phase separation.

Misalnya, trietilamina dan air dapat bercampur dalam semua proporsi di bawah 19 ° C, tetapi di atas suhu kritis ini, solusi komposisi tertentu terpisah menjadi dua fase pada kesetimbangan satu sama lain. Ini berarti bahwa ${\ displaystyle \ Delta _ {mix} G \,}  ÂÂ$ negatif untuk mencampur dua fase di bawah 19 ° C dan positif di atas suhu ini. Oleh karena itu, ${\ displaystyle \ Delta _ {mix} S = - \ left ({\ frac {\ parsial \ Delta _ {campuran} G} {\ parsial T}} \ kanan) _ {P}}  ÂÂ$ negatif untuk mencampur dua fase kesetimbangan ini. Hal ini disebabkan oleh pembentukan ikatan hidrogen yang menarik antara dua komponen yang mencegah pencampuran acak. Molekul Triethylamine tidak dapat membentuk ikatan hidrogen satu sama lain tetapi hanya dengan molekul air, sehingga dalam larutan mereka tetap berhubungan dengan molekul air dengan hilangnya entropi. Pencampuran yang terjadi di bawah 19 ° C disebabkan bukan karena entropi tetapi untuk entalpi pembentukan ikatan hidrogen.

The lower critical solution temperatures also occur in many solvent-polymer blends. For polar systems such as polyacrylic acid in 1,4-dioxane, this is often caused by the formation of hydrogen bonds between the polymer and the solvent. For nonpolar systems such as polystyrene in cyclohexane, phase separation has been observed in closed tubes (at high pressure) at temperatures near the critical point of liquid-vapor of the solvent. At such temperatures, the solvent expands much faster than the polymer, whose segments are covalently connected. Therefore mixing requires solvent contraction for polymer compatibility, resulting in loss of entropy.

src: cf.ppt-online.org

Explanation of statistical thermodynamics of ideal gas mixing entropy

Since thermodynamic entropy can be associated with statistical mechanics or information theory, it is possible to compute entropy mixing using these two approaches. Here we consider a simple case of ideal gas mixing.

Evidence from statistical mechanics

Assume that the molecules of two different substances are approximately the same size, and consider the spaces as divided into rectangular lattices whose cells are molecular sizes. (In fact, the grille will do everything, including close packing.) This is a conceptual model like a crystal to identify the center of a mass molecule. If two phases are fluid, there is no spatial uncertainty in each of them individually. (This is, of course, approximate.The liquid has "free volume".This is why they are (usually) less dense than solid.) Everywhere we look in component 1, there are molecules present, and also for component 2 After two different substances are mixed (assuming they can mix), the liquid is still solid with molecules, but now there is uncertainty about what kind of molecule at that location. Of course, the idea of ​​identifying molecules in a particular location is a thought experiment, not something that can be done, but a well-defined uncertainty calculation.

Kita dapat menggunakan persamaan Boltzmann untuk perubahan entropi yang diterapkan pada proses pencampuran

${\ displaystyle \ Delta _ {mix} S = k_ {B} \ ln \ Omega \,}  ÂÂ$

di mana ${\ displaystyle k_ {B} \,}  ÂÂ$ adalah konstanta Boltzmann. Kami kemudian menghitung jumlah cara ${\ displaystyle \ Omega \,}  ÂÂ$ untuk mengatur ${\ displaystyle N_ {1} \,}  ÂÂ$ molekul komponen 1 dan ${\ displaystyle N_ {2} \,}  ÂÂ$ molekul komponen 2 pada kisi, di mana

${\ displaystyle N = N_ {1} N_ {2} \,}  ÂÂ$

adalah jumlah total molekul, dan oleh karena itu jumlah situs kisi. Menghitung jumlah permutasi ${\ displaystyle N \,}  ÂÂ$ objek, koreksi untuk fakta bahwa ${\ displaystyle N_ {1} \,}  ÂÂ$ dari mereka adalah identik satu sama lain, dan juga untuk ${\ displaystyle N_ {2} \,}  ÂÂ$ ,

${\ displaystyle \ Omega = N!/N_ {1}! N_ {2}! \,}  ÂÂ$

Setelah menerapkan aproksimasi Stirling untuk faktorial dari bilangan bulat besar m:

${\ displaystyle \ ln m! = \ jumlah _ {k} \ ln k \ approx \ int _ {1} ^ {m} dk \ ln k = m \ ln mm }  ÂÂ$ ,

hasilnya adalah ${\ displaystyle \ Delta _ {mix} S = -k_ {B} [N_ {1} \ ln (N_ {1}/N) N_ {2} \ ln (N_ {2}/N)] = - k_ {B} N [x_ {1} \ ln x_ {1} x_ {2} \ ln x_ {2}] \,}  ÂÂ$

di mana kami telah memperkenalkan fraksi mol, yang juga merupakan peluang untuk menemukan komponen tertentu di lokasi kisi yang diberikan.

${\ displaystyle x_ {1} = N_ {1}/N = p_ {1} \; \; {\ teks {dan}} \; \; x_ {2} = N_ {2}/N = p_ {2} \,}  ÂÂ$

Karena Boltzmann konstan