Ideal solution

An ideal solution or ideal mixture is a solution that exhibits thermodynamic properties analogous to those of a mixture of ideal gases. The enthalpy of mixing is zero , as is the volume change on mixing. The vapor pressures of all components obey Raoult's law across the entire range of concentrations, and the activity coefficient (which measures deviation from ideality) is equal to one for each component. The concept of an ideal solution is fundamental to both thermodynamics and chemical thermodynamics and their applications, such as the explanation of colligative properties.

An ideal solution or ideal mixture is a solution that exhibits thermodynamic properties analogous to those of a mixture of ideal gases. The enthalpy of mixing is zero , as is the volume change on mixing. The vapor pressures of all components obey Raoult's law across the entire range of concentrations, and the activity coefficient (which measures deviation from ideality) is equal to one for each component. The concept of an ideal solution is fundamental to both thermodynamics and chemical thermodynamics and their applications, such as the explanation of colligative properties. More formally, for a mix of molecules of A and B, the interactions between unlike neighbors (U AB ) and like neighbors U AA and U BB must be of the same average strength, i.e., 2 U AB = U AA + U BB and the longer-range interactions must be nil (or at least indistinguishable). If the molecular forces are the same between AA, AB and BB, i.e., U AB = U AA = U BB , then the solution is automatically ideal. If the molecules are almost identical chemically, e.g., 1-butanol and 2-butanol, then the solution will be almost ideal. Since the interaction energies between A and B are almost equal, it follows that there is only a very small overall energy (enthalpy) change when the substances are mixed. The more dissimilar the nature of A and B, the more strongly the solution is expected to deviate from ideality. Different related definitions of an ideal solution have been proposed. The simplest definition is that an ideal solution is a solution for which each component obeys Raoult's law p_i=x_ip_i^* for all compositions. Here p_i is the vapor pressure of component i above the solution, x_i is its mole fraction and p_i^* is the vapor pressure of the pure substance i at the same temperature. This definition depends on vapor pressure, which is a directly measurable property, at least for volatile components. The thermodynamic properties may then be obtained from the chemical potential μ (which is the partial molar Gibbs energy g) of each component. If the vapor is an ideal gas, \mu(T,p_i) = g(T,p_i)=g^\mathrm{u}(T,p^u)+RT\ln {\frac{p_i}{p^u}}. The reference pressure p^u may be taken as P^o = 1 bar, or as the pressure of the mix, whichever is simpler. On substituting the value of p_i from Raoult's law, \mu(T,p_i) =g^\mathrm{u}(T,p^u)+RT\ln {\frac{p_i^*}{p^u}} + RT\ln x_i =\mu _i^*+ RT\ln x_i. This equation for the chemical potential can be used as an alternate definition for an ideal solution. However, the vapor above the solution may not actually behave as a mixture of ideal gases. Some authors therefore define an ideal solution as one for which each component obeys the fugacity analogue of Raoult's law f_i = x_i f_i^* . Here f_i is the fugacity of component i in solution and f_i^* is the fugacity of i as a pure substance. Since the fugacity is defined by the equation \mu(T,P) = g(T,P)=g^\mathrm{u}(T,p^u)+RT\ln {\frac{f_i}{p^u}} this definition leads to ideal values of the chemical potential and other thermodynamic properties even when the component vapors above the solution are not ideal gases. An equivalent statement uses thermodynamic activity instead of fugacity. \left(\frac{\partial g(T,P)}{\partial P}\right)_{T}=RT\left(\frac{\partial \ln f}{\partial P}\right)_{T}. Since we know from the Gibbs potential equation that: \left(\frac{\partial g(T,P)}{\partial P}\right)_{T}=v with the molar volume v , these last two equations put together give: \left(\frac{\partial \ln f}{\partial P}\right)_{T}=\frac{v}{RT}. Since all this, done as a pure substance, is valid in an ideal mix just adding the subscript i to all the intensive variables and changing v to \bar{v_i} , with optional overbar, standing for partial molar volume: \left(\frac{\partial \ln f_i}{\partial P}\right)_{T,x_i}=\frac{\bar{v_i}}{RT}. Applying the first equation of this section to this last equation we find: v_i^* = \bar{v}_i which means that the partial molar volumes in an ideal mix are independent of composition. Consequently, the total volume is the sum of the volumes of the components in their pure forms: V = \sum_i V_i^*. \frac{g(T,P)-g^\mathrm{gas}(T,p^u)}{RT}=\ln\frac{f}{p^u}. Remembering 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 turn means that \bar{h_i}=h_i^* and that the enthalpy of the mix is equal to the sum of its component enthalpies. Since \bar{u_i}=\bar{h_i}-p\bar{v_i} and u_i^* = h_i^* - p v_i^* , similarly u_i^*=\bar{u_i}. It is also easily verifiable that C_{pi}^*=\bar{C_{pi}}. \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 we find that \Delta g_{i,\mathrm{mix}}=RT\ln x_i. Since the Gibbs free energy per mole of the mixture G_m is G_m = \sum_i x_i{g_i} then \Delta G_\mathrm{m,mix}=RT\sum_i{x_i\ln x_i}. At last we can calculate the molar entropy of mixing since g_i^*=h_i^*-Ts_i^* and \bar{g_i}=\bar{h_i}-T\bar{s_i} \Delta s_{i,\mathrm{mix}}=-R\sum _i \ln x_i \Delta S_\mathrm{m,mix}=-R\sum _i x_i\ln x_i. \Delta G_{\mathrm{m,mix}} = RT \sum_i x_i \ln x_i or for a two-component ideal solution \Delta G_{\mathrm{m,mix}} = RT (x_A \ln x_A + x_B \ln x_B) where m denotes molar, i.e., change in Gibbs free energy per mole of solution, and x_i is the mole fraction of component i . Note that this free energy of mixing is always negative (since each x_i \in [0,1] , each \ln x_i or its limit for x_i \to 0 must be negative (infinite)), i.e., ideal solutions are miscible at any composition and no phase separation will occur. The equation above can be expressed in terms of chemical potentials of the individual components \Delta G_{\mathrm{m,mix}} = \sum_i x_i \Delta\mu_{i,\mathrm{mix}} where \Delta\mu_{i,\mathrm{mix}}=RT\ln x_i is the change in chemical potential of i on mixing. If the chemical potential of pure liquid i is denoted \mu_i^* , then the chemical potential of i in an ideal solution is \mu_i = \mu_i^* + RT \ln x_i. Any component i of an ideal solution obeys Raoult's Law over the entire composition range: \ p_{i}=(p_{i})_\text{pure} x_i where (p_i)_\text{pure} is the equilibrium vapor pressure of pure component i and x_i\, is the mole fraction of component i in solution.

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