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# Raoult's Second law and the freezing temperature.

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

Some motorists, as far as I know, prefer to fill the tank of 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 which are in the same phase, the component with 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, and the particles 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 concentrations 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 crystalisation temperature decrease and steam pressure decrease of 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 nature of the solute, 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 vapour 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 case of two-component solution

The relative decrease in partial vapor pressure of the solvent (A) above the solution does not depend on the nature of the solute 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

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

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

and

Increase of the boiling point of infinitely dilute solutions of non-volatile substances is not dependent on the nature of the solute 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 molar) - is the ratio of the number of solute moles to the weight of the solvent.

If any of the solutions subjects to the laws of ideal solutions at any concentration it's called 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 are not performed, even in the case of an infinitely dilute solutions.

In 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. The thing is that 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, it 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 freezing point of vodka. I have not found anything accurate on the account of vodka freezing and applying the 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 use of calculator. Although, 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 is it. 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 per kilogram of solvent we have 14.49 mole of alcohol.

Then multiplying by a constant, we find cryoscopic freezing temperature change. By reducing the temperature of crystallization (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 of boiling point.

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

Crystallizing solvent (particularly in the beginning of crystallization) contains fewer impurities (dissolved substances) than in the remaining solution. Repeatedly repeating the crystallization of the solution and each time removing impurities enriched residue solution, we can achieve a significant degree of purification (crystallization method). The same thing happens when boiling - steam contains less impurities in comparison to the remaining solution. The resulting vapor is condensed again and again evaporated, achieving cleaning of the impurities (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 measure the temperature of onset crystallization lowering or raising of the initial boiling point of the solution.
Then calculation goes backward of the given above. Based on the obtained temperature difference and the known cryoscopic and ebullioscopic constants of solvent, the molal concentration of the solute in the solution is determined, and thus its molecular mass.

PLANETCALC, Raoult's Second law and the freezing temperature.