Partial Molal Quantities
The properties of a component of a mixture are expressed as partial molal quantities.
The volume of a solution formed by mixing 50 ml of water and 50 ml of ethanol is 95 ml. when 1 gm of magnesium sulphate is added to 100 ml of water the volume of the solution is less, by about 0.001 ml, than that of the water we started with. When acetone and chloroform are mixed, the solution feels appreciably warm. When methanol and carbon tetrachloride are mixed, the solution feels cool. Many real solutions are not ideal. They cannot be described by the simple addition of the properties of the components of the solution.
Partial molal volume: we begin by seeing how we can assign parts of the volume of a binary solution to reach the two components. The data that are generally available and can be used to obtain volume information are the densities of solutions. These are often reported for solutions with various amounts of the minor component, referred to as the solute, in some fixed amount of the major component, called the solvent. These density data can be used to calculate the volume of solutions with a given amount of solvent and solute. Volumes for solutions with a given molalitites m, defined as the number of moles of solute per 100 g of solvent, are illustrated for aqueous solutions of ammonia and for aqueous solutions of magnesium sulphate.
We introduce the symbol nA for the number of moles of solvent and nB for the number of moles of solute. The slopes of curves such as those of the values of the partial derivative (∂_{V}/∂n_{B}) n_{A}, T, P.
Such derivatives with T, P and the amounts of the other reagents held constant, are examples of partial molal quantities. They play a central role in the thermodynamic treatment of solutions.
Partial molal volumes can be deduced for a component of a solution from volume data of the molal activities. If the components are completely miscible to one another, as are water and ethanol, for example, then each can be treated as the solute component, with carrying concentration, and each as the fixed amount solvent. Partial molal volumes of both components can be reduced by approximate treatment of water ethanol solutions.
Partial molal quantities and the total difference: partial molal derivate terms appear as coefficients when a total differential is written. The volume of a two component mixture at fixed temperature and total pressure is a function of the number of moles of the two components. The total differential that describes this dependence is:
dV = (∂_{V}/∂n_{A})n_{B} dn_{A} + (∂_{V}/∂n_{B})n_{A} dn_{B}
The partial derivate coefficients are seen to be partial molal volumes, which of course are dependent on the composition of the solution. One can in fact carry over the notation procedure of special (sans serif) letters for molar quantities and add a bar over the symbol to designate a partial molal quantity. In this way the nature of partial molal derivates can be emphasized. With,
(∂_{V}/∂n_{A}) n_{B} = V_{A} and (∂_{V}/∂n_{B})n_{A} = V_{B}
One can rewrite as:
dV = V_{A} dn_{A} + V_{B} dn_{B}
Solution properties: changes in the properties of a solution are given by the equation mention above. We can expect that the properties, as well as changes in these properties, can be expressed by an equation involving partial molal quantities. Consider a large amount of binary solution made by repeatedly adding n_{A} mol of A and n_{B} mol of B. now to this solution add another n_{A} mol of A and another nB mol of B. this final addition does not change the composition of the increase the volume by an amount n_{A} V_{A} because of the addition of A and by an amount of n_{B} V_{B} because of the addition of B. the volume change for the amount n_{B} V_{B} because of the addition of B. the volume change for the addition of this increment will be the same as that for any other addition. We can, therefore, interpret the volume of a solution made from n_{A} mol of A and n_{B} mol of B as:
V = nA VA + nB VB
Similar expression can be written for the other thermodynamic properties such as enthalpy, entropy, and free energy.
We have come to a generally applicable relation that lets us describe the properties of a solution in terms of the amounts of the components and combinations express the effectiveness contribution of the component in the solution.
Comparison with ideal solution results: specific expressions for the thermodynamic properties of a binary ideal solution were obtained so that we can illustrate our procedures for obtaining the partial molal component properties by deducing the properties of the components of an ideal solution. The result for free energies, enthalpies, entropies and volumes are:
G_{soln} = [n_{A}G˚_{A} + n_{B}G_{B}] + [n_{A} RT In n_{A}/ (n_{A} + n_{B}) + n_{B} RT In n_{B}/ (n_{A }+ n_{B})]
H_{soln }= [n_{A} H˚_{A} + n_{B} H˚_{B}] + n˚_{B} S˚_{B}] – [n_{A} R In n_{A}/ (n_{A} + n_{B}) + n_{B} R In n_{A}/ (n_{A} + n_{B})]
V_{soln} = [n_{A} V˚_{A} + n_{B} V˚_{B}]
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