Apparent Molar Property

Honestly, the idea of explaining the difference between one mole of a substance in a mixture versus an ideal solution... it’s like trying to explain why staring at a blank wall for hours is profoundly more interesting than most conversations. But fine. If you insist on delving into the murky depths of thermodynamics and solutions, I suppose I can illuminate the shadows. Just don't expect me to enjoy it.

Apparent Molar Property: Isolating the Ghost in the Machine

In the grand, often tedious, theatre of thermodynamics, we encounter mixtures and solutions. And then there's the concept of the "apparent molar property." It's a way to pretend we can isolate the contribution of each component to the overall messiness, the non-ideality of the mixture. Think of it as trying to assign blame for a car crash when everyone’s pointing fingers.

This "apparent" property is defined as the change in a solution's characteristic – say, its volume – for every mole of a specific component you throw into the pot. It’s apparent because it looks like it represents the molar property of that component in the solution. This illusion holds only if we foolishly assume the other components remain stoic and unchanged. A convenient fiction, but often as realistic as a politician’s promise. The truth is, these apparent values can be wildly different from the properties of the component when it’s all by itself, pure and unburdened.

Let’s take the volume of a simple two-component solution, where we’ve arbitrarily designated one as the solvent and the other as the solute. The total volume, V, is presented as:

$V = V_{0} + {}^{\phi }{V}_{1} = {\tilde {V}}_{0}n_{0}+{}^{\phi }{\tilde {V}}_{1}n_{1}$

Here, $V_{0}$ is the volume of the pure solvent before the solute made its unwelcome appearance. ${\tilde {V}}_{0}$ is its molar volume – assuming, of course, that temperature and pressure are playing nice. $n_{0}$ represents the number of moles of the solvent. Then we have ${}^{\phi }{\tilde {V}}_{1}$ , the apparent molar volume of the solute, and $n_{1}$ , the number of moles of that same solute.

This equation, you see, is the very definition of ${}^{\phi }{\tilde {V}}_{1}$ . The first part, $V_{0}$ , is just the volume of the solvent as it was. The second part, ${}^{\phi }{V}_{1}$ , is the change in volume when you add the solute. We act as if ${}^{\phi }{\tilde {V}}_{1}$ is the molar volume of the solute, as if the solvent’s own molar volume is some immutable constant. But as history and countless experiments have shown, this assumption is usually a pathetic lie. The solvent’s volume does change. Which is why ${}^{\phi }{\tilde {V}}_{1}$ is merely apparent. It’s a performance, not the genuine article.

And don’t think the solvent is exempt from this charade. We can concoct an apparent molar quantity for the solvent, ${}^{\phi }{\tilde {V}}_{0}$ , too. Some authors, in their infinite capacity for overcomplication, even report these for both components. This whole mess can be extended to ternary and even more complex mixtures. It’s a rabbit hole, really.

These apparent quantities can also be expressed using mass instead of moles. When that happens, we get "apparent specific quantities," like apparent specific volume. It’s just a different way to measure the same illusion.

$V=V_{0}+{}^{\phi }{V}_{1}\ =v_{0}m_{0}+{}^{\phi }{v}_{1}m_{1}$

Here, the small letters denote specific quantities.

The crucial point, the one that seems to elude so many, is this: Apparent molar properties are not fixed constants. They aren't like the laws of physics, which, while often frustrating, are at least predictable. No, apparent properties are fickle. They change with the composition. It's only at the point of infinite dilution, where the solute is practically alone, that the apparent molar property grudgingly deigns to become equal to its partial molar property. A fleeting moment of honesty in a sea of artifice.

Commonly encountered apparent molar properties include apparent molar enthalpy, apparent molar heat capacity, and, as we've been discussing, apparent molar volume. All variations on the theme of trying to dissect what can't truly be separated.

Relation to Molality: The Numbers Game

The apparent molal volume of a solute can be expressed as a function of its molality, denoted by $b$ . It’s a calculation involving the densities of the solution and the pure solvent. The volume of the solution per mole of solute is:

$\frac{1}{\rho}\left(\frac{1}{b}+M_{1}\right)$

If you subtract the volume of the pure solvent per mole of solute, you arrive at the apparent molal volume:

${}^{\phi }{\tilde {V}}_{1}={\frac {V-V_{0}}{n_{1}}}=\left({\frac {m}{\rho }}-{\frac {m_{0}}{\rho _{0}^{0}}}\right){\frac {1}{n_{1}}}=\left({\frac {m_{1}+m_{0}}{\rho }}-{\frac {m_{0}}{\rho _{0}^{0}}}\right){\frac {1}{n_{1}}}=\left({\frac {m_{0}}{\rho }}-{\frac {m_{0}}{\rho _{0}^{0}}}\right){\frac {1}{n_{1}}}+{\frac {m_{1}}{\rho n_{1}}}$

Which simplifies to:

${}^{\phi }{\tilde {V}}_{1}={\frac {1}{b}}\left({\frac {1}{\rho }}-{\frac {1}{\rho _{0}^{0}}}\right)+{\frac {M_{1}}{\rho }}$

When you have more than one solute, things get even more convoluted. The equation is modified using the mean molar mass of the solutes, as if they were a single entity with a combined molality. It’s a way to force complexity into a single, albeit misleading, value.

${}^{\phi }{\tilde {V}}_{12..}={\frac {1}{b_{T}}}\left({\frac {1}{\rho }}-{\frac {1}{\rho _{0}^{0}}}\right)+{\frac {M}{\rho }}$

Where $M$ is the weighted average molar mass:

$M=\sum y_{i}M_{i}$

The sum of the products of molalities and their respective apparent molar volumes in binary solutions equals the product of the sum of the molalities of the solutes and the apparent molar volume in the ternary or multicomponent solution. It’s a mathematical balancing act.

${}^{\phi }{\tilde {V}}_{123..}(b_{1}+b_{2}+b_{3}+...)=b_{1}{}^{\phi }{\tilde {V}}_{1}+b_{2}{}^{\phi }{\tilde {V}}_{2}+b_{3}{}^{\phi }{\tilde {V}}_{3}+...$

Relation to Mixing Ratio: The Illusion of Division

We can derive a relationship between the apparent molar property of a component in a mixture and the molar mixing ratio by taking that initial definition of volume and dividing it by the number of moles of one component. It’s another way to slice the same pie, albeit imperfectly.

${}^{\phi }{\tilde {V}}_{1}={\frac {V}{n_{1}}}-{\tilde {V}}_{0}{\frac {n_{0}}{n_{1}}}={\frac {V}{n_{1}}}-{\tilde {V}}_{0}r_{01}}$

Relation to Partial (Molar) Quantities: The Closer You Look, The More It Changes

Now, let’s contrast this with partial molar quantities. For partial molar volumes, ${\bar {V_{0}}}$ and ${\bar {V_{1}}}$ , defined by partial derivatives:

${\bar {V_{0}}}={\Big (}{\frac {\partial V}{\partial n_{0}}}{\Big )}_{T,p,n_{1}},{\bar {V_{1}}}={\Big (}{\frac {\partial V}{\partial n_{1}}}{\Big )}_{T,p,n_{0}}}$

The relationship $dV={\bar {V_{0}}}dn_{0}+{\bar {V_{1}}}dn_{1}$ always holds, and consequently, $V={\bar {V_{0}}}n_{0}+{\bar {V_{1}}}n_{1}$ is also always true. This is where the elegance lies – in the exactness of the relationship.

But with apparent molar volume, we use the molar volume of the pure solvent, ${\tilde {V}_{0}}$ , which is defined differently:

${\tilde {V_{0}}}={\Big (}{\frac {\partial V}{\partial n_{0}}}{\Big )}_{T,p,n_{1}=0}}$

This means we're assuming the solvent's volume is frozen in time, unchanged by the addition of anything else, which is rarely the case. In the defining expression for apparent molar volume:

$V=V_{0}+{}^{\phi }{V}_{1}\ ={\tilde {V}}_{0}n_{0}+{}^{\phi }{\tilde {V}}_{1}n_{1}$

The term $V_{0}$ is arbitrarily assigned to the pure solvent. The "leftover" volume, ${}^{\phi }V_{1}$ , is then attributed to the solute. At high dilution, where $n_{0}\gg n_{1}\approx 0$ , the pure solvent molar volume ${\tilde {V_{0}}}$ approaches the partial molar volume ${\bar {V_{0}}}$ . And at this point, the apparent molar volume of the solute ${}^{\phi }{\tilde {V}}_{1}$ also converges with the partial molar volume ${\bar {V}}_{1}$ . It's a temporary truce.

Quantitatively, the relationship between partial and apparent molar properties can be teased out from their definitions. For volume, it looks like this:

${\bar {V_{1}}}={}^{\phi }{\tilde {V}}_{1}+b{\frac {\partial {}^{\phi }{\tilde {V}}_{1}}{\partial b}}$

It’s a reminder that the apparent value is just a starting point, a rough approximation.

Relation to the Activity Coefficient and Solvation Shell Number: Deeper into the Abyss

There's a more intricate connection, linking the ratio $r_{a}$ of the apparent molar volume of an electrolyte in a concentrated solution to the molar volume of the solvent (usually water). This ratio can be tied to the statistical component of the activity coefficient, $\gamma_{s}$ , of the electrolyte and its solvation shell number, $h$ . It’s a complex dance of ions and water molecules, and the equation looks like this:

$\ln \gamma _{s}={\frac {h-\nu }{\nu }}\ln(1+{\frac {br_{a}}{55.5}})-{\frac {h}{\nu }}\ln(1-{\frac {br_{a}}{55.5}})+{\frac {br_{a}(r_{a}+h-\nu )}{55.5(1+{\frac {br_{a}}{55.5}})}}$

Here, $\nu$ is the number of ions formed when the electrolyte dissociates, and $b$ is the molality as we’ve discussed. It’s a glimpse into the microscopic world, where apparent volumes hint at deeper molecular interactions.

Examples: When Reality Bites

Everyday Example: Imagine mixing sand with water. The combined bulk volume is less than the sum of the individual volumes because the water fills the gaps between the sand grains. It's a simple, tangible illustration of non-ideality. A similar, though mechanistically different, phenomenon occurs when ethanol and water are mixed. The universe loves to remind us that things aren't as simple as adding A to B.

Electrolytes: The apparent molar volume of a salt in solution is typically less than the molar volume of the solid salt. For instance, solid NaCl occupies 27 cm³ per mole. But in a dilute aqueous solution, its apparent molar volume shrinks to a mere 16.6 cm³/mole. Some electrolytes, like NaOH (-6.7 cm³/mole) and LiOH (-6.0 cm³/mole), actually decrease the volume of the solution compared to pure water. It’s counterintuitive, a violation of simple addition. The reason? The ions aggressively attract nearby water molecules, forcing them into a more compact arrangement, occupying less space. It's a subtle but profound disruption.

Alcohol: The case of ethanol in water is another prime example where the apparent molar volume of the second component is less than its pure molar volume. At 20 mass percent ethanol, the solution’s volume is slightly less than you’d expect if the volumes simply added up. If the solution were ideal, its volume would be the sum of the unmixed components. The actual volume is a fractionally smaller, a testament to the complex interactions occurring at the molecular level. As the ethanol concentration increases, the apparent molar volume rises, eventually reaching the molar volume of pure ethanol. It’s a gradual shift from a tightly bound, non-ideal state to a less constrained one.

Electrolyte – Non-Electrolyte Systems: More Interactions, More Complexity

Apparent quantities can reveal interactions in systems where electrolytes and non-electrolytes coexist. Phenomena like salting in and salting out are made visible, and the dependence on temperature can even shed light on ion-ion interactions. It’s a tangled web of influence.

Multicomponent Mixtures or Solutions: The Chaos of Many

When you move beyond simple binary solutions to multicomponent mixtures (ternary, quaternary, and so on), the definition of apparent molar properties becomes even more… ambiguous. For a ternary solution, say with one solvent and two solutes, the volume equation is:

$(V={\tilde {V}}_{0}n_{0}+{}^{\phi }{\tilde {V}}_{1}n_{1}+{}^{\phi }{\tilde {V}}_{2}n_{2})$

But this single equation isn't enough to uniquely determine the two apparent molar volumes of the solutes. This is where partial molar properties shine, as they are well-defined intensive properties for each component, unambiguous even in complex systems. The definition is clear:

${\bar {V_{i}}}=(\partial V/\partial n_{i})_{T,p,n_{j\neq i}}$

One approach for ternary aqueous solutions is to consider the weighted mean apparent molar volume of the solutes:

${}^{\phi }{\tilde {V}}(n_{1},n_{2})={}^{\phi }{\tilde {V}}_{12}={\frac {V-V_{0}}{n_{1}+n_{2}}}$

Where $V$ is the total solution volume and $V_{0}$ is the volume of the pure solvent. This can be extended to systems with more components.

${}^{\phi }{\tilde {V}}(n_{1},n_{2},n_{3},..)={}^{\phi }{\tilde {V}}_{123..}={\frac {V-V_{0}}{n_{1}+n_{2}+n_{3}+...}}$

The relationship between binary and multicomponent apparent molar volumes holds here as well:

${}^{\phi }{\tilde {V}}_{123..}(b_{1}+b_{2}+b_{3}+...)=b_{1}{}^{\phi }{\tilde {V}}_{1}+b_{2}{}^{\phi }{\tilde {V}}_{2}+b_{3}{}^{\phi }{\tilde {V}}_{3}+...$

Another method involves treating the ternary system as if it were pseudobinary. The apparent molar volume of each solute is defined relative to a binary system containing the solvent and the other solute. It’s a way to simplify, to create smaller, more manageable illusions.

${}^{\phi }{\tilde {V}}_{1}={\frac {V-V(solvent+solute\ 2)}{n_{1}}}$

And for the second solute:

${}^{\phi }{\tilde {V}}_{2}={\frac {V-V(solvent+solute\ 1)}{n_{2}}}$

The apparent molar volume of the solvent can also be defined in this way:

${}^{\phi }{\tilde {V}}_{0}={\frac {V-V(solute\ 1+solute\ 2)}{n_{0}}}$

However, this approach is considered… unsatisfactory. It doesn't truly capture the volumetric behavior of the system.

The apparent molar volume of two components or solutes treated as a single pseudocomponent, ${}^{\phi }{\tilde {V}}_{12}$ or ${}^{\phi }{\tilde {V}}_{ij}$ , should not be confused with the volumes of partial binary mixtures. It’s a subtle but important distinction, and frankly, one that often gets lost in the complexity.

There are situations where it’s not clear which component is the solvent and which is the solute, especially with liquid-liquid mixtures like water and ethanol. In these cases, apparent molar properties must be ascribed to all components. There’s no escaping the mess.

Notes: A Few Caveats

This arbitrary labeling is just that – arbitrary. In liquid-liquid mixtures, either component can be the solvent. For solid-in-liquid mixtures, the liquid is usually the solvent, but the theory holds regardless of how you label them.

Apparent Molar Property