# Raoult's Second law and the freezing temperature.

Calculation of boiling point and crystallization (freezing) of non-electrolyte solutions.

As far as I know, some motorists prefer to fill the tank of a screen wiper with vodka in the winter. Vodka does not freeze in the winter, but why and up to what temperature? The chemistry will give us an answer.

As we know, vodka is a solution of ethyl alcohol in water.

And what is the solution? It is a homogeneous mixture of at least two components, one of which is called the solvent and another solute. The solvent is a component, which an aggregate state of which is not changed during the formation of solution (e.g., sugar in water changes from solid to liquid, water is a solvent) or, in the case of substances that are in the same phase, the component with the highest amount. The solutions can be solid, liquid, and gaseous (air as a mixture gasses is a gaseous solution).

From the chemical point of view, the solution is a dispersion system, i.e., a system where two or more substances are in a crushed state. The particles are evenly distributed relative to each other and interact.

The difference here is in the degree of dispersion.
If the particle size of the substances that make up the system are equal to or less than
$10^{-7}$ (the size of atoms, molecules and ions then it's a molecular dispersion system or molecular solution`.
If the particle size of the substances are $10^{-5}-10^{-7}$ then it's a colloid-dispersed system or colloid solution.
If the particle size is more than $10^{-5}$ then it's a coarser-grained system.

Among the true solutions, there are 2 classes - electrolyte solutions (ions), which conduct electric current, and non-electrolyte solutions (molecules).

Particles mixed in a solution can interact with each other. Due to the presence or absence of interaction of solution particles with each other, they can be put in two groups - real and ideal.
The properties of initial molecules are changed due to the intermolecular and chemical interaction of particles in real solutions. In ideal solutions, there is practically no interaction between the particles, and the solute retains its properties. The ideal solutions at any concentration are those solutions, the components of which are very similar in physical and chemical properties and the formation of which is not accompanied by a change in volume and the release or absorption of heat.
In 1887, French chemist François-Marie Raoult studied crystallization temperature decrease and steam pressure decrease of a solvent when introduced to solute and discovered a series of laws known as Raoult's laws. These are quantitative patterns, describing colligative, i.e., depending on the concentration but not on the solute's nature, properties of the solutions. These laws and describe the behavior of ideal solutions.

Raoult's First law states(see Raoult's law) that partial pressure of vapor is proportionate to its mole fraction in the solution, with a proportionality coefficient equal to the saturated vapor pressure over the pure component.
$P_i=P_i^0X_i$

Or in the case of a two-component solution

The relative decrease in partial vapor pressure of the solvent (A) above the solution does not depend on the solute's nature and is equal to its mole fraction in the solution.

$\frac{(P_A^0-P_A)}{P_A^0}=X_B$

This law has two consequences, which are called Raoult's Second law.

Raoult's Second law states that

A decrease in crystallization temperature of infinitely dilute solutions does not depend on the solute's nature and directly proportional to the molal concentration of the solution.

$T_{fr}^0-T_{fr}=\Delta T_{fr}=Km$

and

An increase of the boiling point of infinitely dilute solutions of non-volatile substances is not dependent on the solute's nature and directly proportional to the molal concentration of the solution.

$T_{b}^0-T_{b}=\Delta T_{b}=Em$

The proportionality factors K and E in these equations are - cryoscopic and boiling constants of the solvent, having a physical meaning of the crystallization temperature and increase of the solution boiling point with the molal concentration of 1 mol/kg. Solutions with such concentration -
Растворы с такой концентрацией - 1 mol/kg, generally speaking, can not be called an infinitely diluted, so that the determination of these constants we are talking about dependence extrapolation of low concentrations. Remind that the molal concentration (as opposed to a molar) - is the ratio of the number of solute moles to the solvent's weight.

If any of the solutions subject to the laws of ideal solutions at any concentration, it's called a perfect solution. If it subjects to the laws at sufficiently large dilution, it's infinitely dilute solution.

All the electrolyte solutions - real solutions, as solute therein dissociates into ions. Raoult's law for these solutions is not performed, even in the case of an infinitely dilute solution.

In the case of non-electrolyte solutions - the more dilute the solution, the closer its properties ideal. Homogeneous mixtures of non-polar substances (hydrocarbons) are close to the ideal solution at all concentrations.
Now let's get back to vodka.
----------------------Update------------------------
So, thanks to inquisitive users (see the comments to the calculator), the author had to find out that Raoult's second law has nothing to do with vodka. In Raoult's laws, we are talking about solutions of non-volatile matter (like salt, for example), which reduce the vapor pressure of the solvent above the solution, and alcohol - quite a volatile substance, also creates a vapor pressure above the solution. For boiling vodka, there are Konovalov laws applicable, and alcohol from vodka starts to boil out at the boiling point of alcohol (as I understand).
However, in several places on the Internet, I've seen the use of Raoult's second law to estimate the freezing point of vodka. I have not found anything accurate on vodka freezing and applying Raoult's second law to this(I need a chemist). However, the findings are quite close to tabular, so I leave the whole calculation below unchanged to illustrate the calculator's use with a proviso that the boiling temperature and, probably, freezing temperature cannot be determined with Raoult's second law.
---------------------End of update-----------------------

Vodka is a hydrocarbon dissolved in water. Therefore, apply the second law of Raul to determine the freezing temperature of vodka.

The solvent, in this case, is water. Cryoscopic constant and ebullioscopic constants for it are given in reference /350/. The percentage of alcohol and water is known - 40%. From this, we can determine the molal concentration of vodka.

Define how much alcohol (m1) should be added per kilogram of water (m2), to obtain a 40% ratio (K)
$\frac{m_1}{m_1 + m_2}=K$,
therefore
$m_1 = \frac{Km_2}{1 - K}$

Thus, to achieve a 40% solution in 1 kg of water, it's necessary to pour about 666.6 (6) grams of alcohol.

Now we have to determine how many moles it is. For this, we need to know the molar mass of alcohol. Given the fact that the formula of ethanol is known to all $C_2H_5OH$, then use the calculator Molar mass of the substance, we find that the molar mass of alcohol 46 g / mol. Dividing the weight of alcohol by its molar mass, we find that we have 14.49 mole of alcohol per kilogram of solvent.

Then multiplying by a constant, we find cryoscopic freezing temperature change. By reducing the crystallization temperature (freezing) of the solvent - water, we will find the crystallization temperature (-27) vodka.

However, as applied to the solutions, people do not say "solution crystallization temperature" and "boiling point of the solution." They say so - "onset temperature of crystallization" and "initial boiling point".

The fact that both during boiling (the solvent evaporates) and during the crystallization (solvent crystals evolve), solute concentration increase, and there is a further reduction in the crystallization temperature or an increase in boiling point.

The methods of cleaning substances are based on this effect, i.e., cleaning of solvent, e.g., water from impurities that can not be removed by conventional filtration.

Crystallizing solvent (particularly at the beginning of crystallization) contains fewer impurities (dissolved substances) than in the remaining solution. Repeatedly repeating the solution's crystallization and removing impurities enriched residue solution, we can achieve a significant degree of purification (crystallization method). The same thing happens when boiling - steam contains fewer impurities in comparison to the remaining solution. The resulting vapor is condensed again and again evaporated, achieving the impurities' cleaning (distillation method).

There is a calculator to determine the initial freezing and boiling solution below.
The default values match vodka.

#### Initial boiling point and crystallization (freezing) of non-electrolyte solutions

Digits after the decimal point: 1
Crystallization temperature (freezing)

Boiling temperature

By the way, Raoult's second law is also used for the experimental determination of the molar mass of unknown substances. For this purpose, a mass of the test substance is dissolved in an appropriate solvent and measures the temperature of onset crystallization, lowering or raising the solution's initial boiling point.
Then calculation goes backward of the given above. Based on the obtained temperature difference and the known cryoscopic and ebullioscopic constants of solvent, the solute's molal concentration in the solution is determined, and thus its molecular mass.

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