Clapeyron's equation

When integrating the Clausius-Clapeyron equation in the simplest case, it is assumed that the group has a constant value that does not depend on temperature.

Denoting the constant value of integration by “A”, we obtain

Relation (7.5) is sometimes called the Clapeyron equation. Graphically, the given dependence is expressed by a straight line. Equation (7.5) often turns out to be a good approximation, but in the general case it gives significant errors due to the fact that the dependence on has S-shaped view. Equation (7.5) is not applicable for temperatures below the normal boiling point, even for non-polar and non-associative substances. For the last linear form P-T dependencies can be used to approximate P-T data only in a narrow temperature range, even in the temperature range above the normal boiling point.

Antoine correlation for vapor pressure

Antoine proposed a widely used simple modification of equation (7.5):

At C=0 equation (7.6) turns into the Clapeyron equation (7.5). The constants “A”, “B” and “C” in the Antoine equation are obtained by approximating the experimental P-T data. For many substances, the values ​​of these constants are given in the reference literature. The applicability of the Antoine equation constants is limited to those temperature or pressure ranges for which they are recommended by the authors of these reference books. You cannot use the Antoine equation outside the recommended intervals.

Cox-Antoine correlation for vapor pressure

Cox proposed a graphical vapor pressure correlation. On the graph, the logarithm is plotted along the ordinate axis and a straight line (with a positive slope) is drawn, the ordinates of which correspond to the vapor pressure of water (or some other reference substance). Since the dependence of water vapor pressure on temperature is well known, the abscissa is accordingly marked in units of temperature. On the coordinate field prepared in this way, the vapor pressures of other substances will also be shown as straight lines. However, such graphs are somewhat inconvenient for practical use due to the fact that the temperature interpolation turns out to be very inaccurate.

Kalingert and Davis showed that the temperature scale thus obtained is almost equivalent to a function; here "C" is approximately equal to 43 K for most substances boiling at temperatures from 0 to 100 C. The same authors constructed the vapor pressure dependences for a number of substances in the indicated way and obtained almost straight lines. Their equation

very similar to the Antoine equation, is often called just that, and its graphic representation is considered a Cox diagram.

There are indications in the literature that there are quite simple rules, linking the constant "C" with the normal boiling point of the substance. Although these rules are not sufficiently reliable, some of them are given below in order to familiarize yourself with the approaches.

Thompson Rules:

for monatomic and all substances with a normal boiling point K

for other compounds

For substances whose normal boiling point is above 250 K, it is recommended to take C = 43 K;

for low-boiling gases C 0.

Another, more common form of the Cox-Antoine correlation is obtained by differentiating equation (7.7) with respect to 1/T and combining the obtained dependence with the Clausius-Clapeyron equation (7.3). For the normal boiling point, the constants “A” and “B” are expressed in this case as follows:

Where P vp expressed in physical atmospheres.

To use equations (7.8), it is only necessary to know at T b And T b. . In accordance with Miller's recommendation, for most substances in the low pressure region, ~ 1.05 can be accepted.

I once made pancakes in a group of comrades, where teetotalers-ulcers were present. I’m lying, there didn’t seem to be ulcers, but teetotalers were special. Some flatly refused to consume ethyl alcohol in any form, even in the form of kefir. And arguments like the fact that the human body itself produces alcohol in a certain amount did not pass. It works out - let it, but this is an unconscious process (read irrational), and we, as the embodiment of rationality, do not want to add to it what we do not want to add.

Yes, I forgot to say, I wanted to cook pancakes not on anything, but on real beer - 5% of this very pentahydrodicarbonium hydroxide of ethyl alcohol.

I had to think of other rational arguments.

It was necessary to somehow demonstrate (read prove) the absence of alcohol in the finished product.

Well, let's remember our youth.

The mass of a substance evaporated from any surface is calculated by the formula (we remember this; and everything is logical):

m = W * S * t, where
W - evaporation rate,
S - surface area in m²,
t - time in s.

We know the mass (neglecting the density): 0.5 kg 5% - 25 g = 0.025 kg of alcohol. But this is in the whole pancake mass. Considering that I planned to get about 40 pancakes out of the volume of dough for which one bottle of beer goes, which means that there are ~ 0.000625 kg of alcohol per pancake. A little. But digital techies require arguments only in the form of figures, sometimes condescending to graphs and diagrams. OK. Let's continue.

We also know the evaporation surface area - let's take it as the pancake area (i.e. pans 22 cm = 0.22 m) = π * 0.22² ~ 0.1521 m²

Now you need to find out the rate of evaporation of alcohol.
Then I had to get on the Internet, which said that the evaporation rate is calculated as
W = 10⁻⁶ * n * √m * P,
where n is a coefficient that takes into account the speed air flow environment(taken from tables). In this case, I decided to neglect it, in the sense of taking it as 1 (ie, flow = 0 m/s).
m is the molecular weight of the substance. Oh, this is easy enough. C₂H₅OH - 46.07 g / mol (the Internet helped us here).
But P is the saturated vapor pressure of a substance at a specific temperature and is calculated using the Antoine equation
lgP \u003d A-B / (C + T), where T is the design temperature, and A, B, C are the Antoine equation constants for the dependence of saturated vapor pressure on temperature.

Eprsh. Normal people take such values ​​​​from reference books, but, as luck would have it, I didn’t have anything suitable at hand. And the Internet, the infection, was silent. Apparently, he asked the wrong questions in the park ...

But ... the baking temperature of pancakes is clearly higher than the boiling point of alcohol (would not confuse which of them does what), and this cannot mean anything else, except that the pressure of the evaporated substance is equal to the external pressure, i.e. let Antoine nervously smoke on the sidelines, we will take the standard atmospheric pressure data - 100 kPa.

So, all the initial data is there. We believe:

W = 10⁻⁶ * 1* √46.07 * 100 = 0.00068 kg/m²s
t = 0.000625 / (0.00068 * 0.1521) = 6.04 s

Get it. All the alcohol will evaporate from the pancake in less than 10 seconds.

The teetotalers had to check the calculations, grumble about the incorrectness of the applied method (and the chemical activity of the substance was not taken into account and the calculations, they say, were carried out for a pure substance, etc.), but they still ate the pancakes. Q.E.D!

Construction of a phase equilibrium diagram for a binary mixture

1. Task content:

Formulation of the problem:

At pressure P, construct a phase equilibrium diagram for a given 2-component mixture using the following models: a) ideal mixture; b) Wilson; c) NRTL.

Given:

P, substances.

Build:

I schedule: T = f (x A); T = f (y A) - the number of points N = 101 of the mixture according to the models: a) ideal mixture; b) Wilson; c) NRTL.

II schedule: y A = f (x A) - the number of points N = 101. mixtures by models: a) ideal mixture; b) Wilson; c) NRTL.

Task options:

	Components				Components

Legend:

A– volatile component;

B– hardly volatile component;

x i- the content of the i-th component in the liquid phase, (mol.%);

y i- the content of the i-th component in the vapor phase, (mol.%);

i– component index;

k i is the phase equilibrium constant of the i-th component;

P– pressure in the system, ata;

Pi is the vapor pressure over the pure solvent, atm;

T– system temperature, °C

T balei– boiling point of the i-th component, °C

γ - activity coefficient

Λ ij, Vi, λij are the parameters of the Wilson model;

Gij, g ij are the parameters of the NRLT model;

Mathematical model:

Stoichiometric ratio:

∑x i = 1; ∑y i = 1

Phase balance:

Phase equilibrium constant:

Saturated vapor pressure of the i-th component:

Activity coefficients:

a) the perfect mixture

b) according to Wilson

Note: the superscript is a square.

Reference data:

Antoine equation coefficient

V1 = 104.00; V2 = 49.60; V 3 \u003d 18.70

a 12 = 0.30; a 13 = 0.20; a23 = 0.46

Calculation algorithm for the ideal mixture model:

1. Write out coefficients Ai ,B i ,C i Antoine equations for a given pair of substances.

2. Find the boiling points of substances according to the "boiling point search algorithm T bale at a given pressure of the system P "and determine from a pair of substances a highly volatile substance A and a heavily volatile substance B.

3. Find the temperature step

4. We find Tj at j = 1, … , N.

T1 =T HT j +1 =T j +Δ T

5. For each Tj find P A And PB according to Antoine's equation.

6. For each Tj find K A And K B at γ = 1.

7. For each Tj find x A

8. For each Tj find y A.

9. We build graphs.

Calculation algorithm for Wilson models andNRLT:

items 1-7 are the same as in the "calculation algorithm according to the model of an ideal mixture"

8. For x 1 =x A And x 2 \u003d 1 -x A find the natural logarithms of the activity coefficients ln γ 1 and ln γ 2 according to the Wilson model or NRLT.

9. Find activity coefficients γ 1 And γ 2 according to the Wilson model or NRLT.

10. For each Tj find K A And K B at γ 1 And γ 2, calculated in item 9.

11. For each Tj clarify x A

12. For each Tj find y A.

13. We build graphs.

Boiling temperature search algorithmT bale at system pressureP:

1. Set an arbitrary temperature T.

2. Find Pi given substance at a given temperature T according to Antoine's equation.

3. If | Pi – P|< 0.001 then T bale = T. If | Pi – P| ≥ 0.001, then go to item 1, choosing T until the condition of item 3 is satisfied.

2. Report content:

Formulation of the problem

Goal of the work

The progress of the work, with a description of the calculations, the results of which are presented in the form of graphs;

3. Questions to control:

1) The main stages of constructing a mathematical description of mass transfer processes. What is the mathematical description of the mass transfer process based on?

2) The physical meaning of the phase equilibrium diagram. Heterogeneous and homogeneous systems. Dependence of phase equilibrium diagrams on pressure.

3) The fugacity of the component in the mixture, the activity coefficient of the component.

4) Wilson's equation (the concept of local compositions). NRTL equation (2 sorts of cells).