The distance between molecules in a solid. Ideal gas. Parameters of the ideal gas state. Gas laws and fundamentals of the ICT

Molecules are very small, ordinary molecules cannot be seen even with the most powerful optical microscope - but some parameters of molecules can be calculated quite accurately (mass), and some can only be very roughly estimated (dimensions, speed), and it would also be good to understand what “size” is molecules" and what kind of "molecule speed" we are talking about. So, the mass of a molecule is found as “the mass of one mole” / “the number of molecules in a mole”. For example, for a water molecule m = 0.018/6·1023 = 3·10-26 kg (you can calculate more precisely - Avogadro’s number is known with good accuracy, and the molar mass of any molecule is easy to find).
Estimating the size of a molecule begins with the question of what constitutes its size. If only she were a perfectly polished cube! However, it is neither a cube nor a ball, and in general it does not have clearly defined boundaries. What to do in such cases? Let's start from a distance. Let's estimate the size of a much more familiar object - a schoolchild. We have all seen schoolchildren, let’s take the mass of an average schoolchild to be 60 kg (and then we’ll see whether this choice has a significant effect on the result), the density of a schoolchild is approximately like that of water (remember that if you take a deep breath of air, and after that you can “hang” in the water, immersed almost completely, and if you exhale, you immediately begin to drown). Now you can find the volume of a schoolchild: V = 60/1000 = 0.06 cubic meters. meters. If we now assume that the student has the shape of a cube, then its size is found as the cube root of the volume, i.e. approximately 0.4 m. This is how the size turned out - less than the height (the “height” size), more than the thickness (the “depth” size). If we don’t know anything about the shape of a schoolchild’s body, then we won’t find anything better than this answer (instead of a cube we could take a ball, but the answer would be approximately the same, and calculating the diameter of a ball is more difficult than the edge of a cube). But if we have additional information (from analysis of photographs, for example), then the answer can be made much more reasonable. Let it be known that the “width” of a schoolchild is on average four times less than his height, and his “depth” is three times less. Then Н*Н/4*Н/12 = V, hence Н = 1.5 m (there is no point in making a more accurate calculation of such a poorly defined value; relying on the capabilities of a calculator in such a “calculation” is simply illiterate!). We received a completely reasonable estimate of the height of a schoolchild; if we took a mass of about 100 kg (and there are such schoolchildren!), we would get approximately 1.7 - 1.8 m - also quite reasonable.
Let us now estimate the size of a water molecule. Let’s find the volume per molecule in “liquid water” - in it the molecules are most densely packed (pressed closer to each other than in the solid, “ice” state). A mole of water has a mass of 18 g and a volume of 18 cubic meters. centimeters. Then the volume per molecule is V= 18·10-6/6·1023 = 3·10-29 m3. If we do not have information about the shape of a water molecule (or if we do not want to take into account the complex shape of molecules), the easiest way is to consider it a cube and find the size exactly as we just found the size of a cubic schoolchild: d= (V)1/3 = 3·10-10 m. That's all! You can evaluate the influence of the shape of fairly complex molecules on the calculation result, for example, like this: calculate the size of gasoline molecules, counting the molecules as cubes - and then conduct an experiment by looking at the area of ​​the spot from a drop of gasoline on the surface of the water. Considering the film to be a “liquid surface one molecule thick” and knowing the mass of the drop, we can compare the sizes obtained by these two methods. The result will be very instructive!
The idea used is also suitable for a completely different calculation. Let us estimate the average distance between neighboring molecules of a rarefied gas for a specific case - nitrogen at a pressure of 1 atm and a temperature of 300 K. To do this, let’s find the volume per molecule in this gas, and then everything will turn out simple. So, let’s take a mole of nitrogen under these conditions and find the volume of the portion indicated in the condition, and then divide this volume by the number of molecules: V= R·T/P·NA= 8.3·300/105·6·1023 = 4·10 -26 m3. Let us assume that the volume is divided into densely packed cubic cells, and each molecule “on average” sits in the center of its cell. Then the average distance between neighboring (closest) molecules is equal to the edge of the cubic cell: d = (V)1/3 = 3·10-9 m. It can be seen that the gas is rarefied - with such a relationship between the size of the molecule and the distance between the “neighbors” the molecules themselves occupy a rather small - approximately 1/1000 part - of the volume of the vessel. In this case, too, we carried out the calculation very approximately - there is no point in calculating such not very definite quantities as “the average distance between neighboring molecules” more accurately.

Gas laws and fundamentals of the ICT.

If the gas is sufficiently rarefied (and this is a common thing; we most often have to deal with rarefied gases), then almost any calculation is made using a formula connecting pressure P, volume V, amount of gas ν and temperature T - this is the famous “equation state of an ideal gas" P·V= ν·R·T. How to find one of these quantities if all the others are given is quite simple and understandable. But the problem can be formulated in such a way that the question will be about some other quantity - for example, about the density of a gas. So, the task: find the density of nitrogen at a temperature of 300K and a pressure of 0.2 atm. Let's solve it. Judging by the condition, the gas is quite rarefied (air consisting of 80% nitrogen and at significantly higher pressure can be considered rarefied, we breathe it freely and easily pass through it), and if this were not so, we don’t have any other formulas no – we use this favorite one. The condition does not specify the volume of any portion of gas; we will specify it ourselves. Let's take 1 cubic meter of nitrogen and find the amount of gas in this volume. Knowing the molar mass of nitrogen M = 0.028 kg/mol, we find the mass of this portion - and the problem is solved. Amount of gas ν= P·V/R·T, mass m = ν·М = М·P·V/R·T, hence density ρ= m/V = М·P/R·T = 0.028·20000/( 8.3·300) ≈ 0.2 kg/m3. The volume we chose was not included in the answer; we chose it for specificity - it’s easier to reason this way, because you don’t necessarily immediately realize that the volume can be anything, but the density will be the same. However, one can figure out: “by taking a volume, say, five times larger, we will increase the amount of gas exactly five times, therefore, no matter what volume we take, the density will be the same.” You could simply rewrite your favorite formula, substituting into it the expression for the amount of gas through the mass of a portion of gas and its molar mass: ν = m/M, then the ratio m/V = M P/R T is immediately expressed, and this is the density . It was possible to take a mole of gas and find the volume it occupies, after which the density is immediately found, because the mass of the mole is known. In general, the simpler the problem, the more equivalent and beautiful ways to solve it...
Here is another problem where the question may seem unexpected: find the difference in air pressure at a height of 20 m and at a height of 50 m above ground level. Temperature 00C, pressure 1 atm. Solution: if we find the air density ρ under these conditions, then the pressure difference ∆P = ρ·g·∆H. We find the density in the same way as in the previous problem, the only difficulty is that air is a mixture of gases. Assuming that it consists of 80% nitrogen and 20% oxygen, we find the mass of a mole of the mixture: m = 0.8 0.028 + 0.2 0.032 ≈ 0.029 kg. The volume occupied by this mole is V= R·T/P and the density is found as the ratio of these two quantities. Then everything is clear, the answer will be approximately 35 Pa.
The gas density will also have to be calculated when finding, for example, the lifting force of a balloon of a given volume, when calculating the amount of air in scuba cylinders required for breathing under water for a certain time, when calculating the number of donkeys required to transport a given amount of mercury vapor through the desert and in many other cases.
But the task is more complicated: an electric kettle is boiling noisily on the table, the power consumption is 1000 W, efficiency. heater 75% (the rest “goes” into the surrounding space). A jet of steam flies out of the spout - the area of ​​the “spout” is 1 cm2. Estimate the speed of the gas in this jet. Take all the necessary data from the tables.
Solution. Let's assume that saturated steam is formed above the water in the kettle, then a stream of saturated water vapor flies out of the spout at +1000C. The pressure of such steam is 1 atm, it is easy to find its density. Knowing the power used for evaporation Р= 0.75·Р0 = 750 W and the specific heat of vaporization (evaporation) r = 2300 kJ/kg, we will find the mass of steam formed during time τ: m= 0.75Р0·τ/r. We know the density, then it is easy to find the volume of this amount of steam. The rest is already clear - imagine this volume in the form of a column with a cross-sectional area of ​​1 cm2, the length of this column divided by τ will give us the speed of departure (this length takes off in a second). So, the speed of the jet leaving the spout of the kettle is V = m/(ρ S τ) = 0.75 P0 τ/(r ρ S τ) = 0.75 P0 R T/(r P M ·S) = 750·8.3·373/(2.3·106·1·105·0.018·1·10-4) ≈ 5 m/s.
(c) Zilberman A.R.

Let us consider how the projection of the resulting force of interaction between them on the straight line connecting the centers of the molecules changes depending on the distance between the molecules. If molecules are located at distances several times greater than their sizes, then the interaction forces between them have practically no effect. The forces of interaction between molecules are short-range.

At distances exceeding 2-3 molecular diameters, the repulsive force is practically zero. Only the force of attraction is noticeable. As the distance decreases, the force of attraction increases and at the same time the force of repulsion begins to affect. This force increases very quickly when the electron shells of the molecules begin to overlap.

Figure 2.10 graphically shows the projection dependence F r the forces of interaction of molecules on the distance between their centers. On distance r 0, approximately equal to the sum of the molecular radii, F r = 0 , since the force of attraction is equal in magnitude to the force of repulsion. At r > r 0 there is an attractive force between the molecules. The projection of the force acting on the right molecule is negative. At r < r 0 there is a repulsive force with a positive projection value F r .

Origin of elastic forces

The dependence of the interaction forces between molecules on the distance between them explains the appearance of elastic force during compression and stretching of bodies. If you try to bring the molecules closer to a distance less than r0, then a force begins to act that prevents the approach. On the contrary, when molecules move away from each other, an attractive force acts, returning the molecules to their original positions after the cessation of external influence.

With a small displacement of molecules from equilibrium positions, the forces of attraction or repulsion increase linearly with increasing displacement. In a small area, the curve can be considered a straight segment (the thickened section of the curve in Fig. 2.10). That is why, at small deformations, Hooke’s law turns out to be valid, according to which the elastic force is proportional to the deformation. At large molecular displacements, Hooke's law is no longer valid.

Since the distances between all molecules change when a body is deformed, the neighboring layers of molecules account for an insignificant part of the total deformation. Therefore, Hooke's law is satisfied at deformations millions of times greater than the size of the molecules.

Atomic force microscope

The device of an atomic force microscope (AFM) is based on the action of repulsive forces between atoms and molecules at short distances. This microscope, unlike a tunnel microscope, allows you to obtain images of surfaces that do not conduct electrical current. Instead of a tungsten tip, AFM uses a small fragment of diamond, sharpened to atomic size. This fragment is fixed on a thin metal holder. As the tip approaches the surface under study, the electron clouds of diamond and surface atoms begin to overlap and repulsive forces arise. These forces deflect the tip of the diamond tip. The deviation is recorded using a laser beam reflected from a mirror mounted on a holder. The reflected beam drives a piezoelectric manipulator, similar to the manipulator of a tunnel microscope. The feedback mechanism ensures that the height of the diamond needle above the surface is such that the bend of the holder plate remains unchanged.

In Figure 2.11 you see an AFM image of the polymer chains of the amino acid alanine. Each tubercle represents one amino acid molecule.

At present, atomic microscopes have been constructed, the design of which is based on the action of molecular forces of attraction at distances several times greater than the size of an atom. These forces are approximately 1000 times less than the repulsive forces in AFM. Therefore, a more complex sensing system is used to record the forces.

Atoms and molecules are made up of electrically charged particles. Due to the action of electrical forces over short distances, molecules are attracted, but begin to repel when the electron shells of the atoms overlap.

An example of the simplest system studied in molecular physics is gas. According to the statistical approach, gases are considered as systems consisting of a very large number of particles (up to 10 26 m –3) that are in constant random motion. In molecular kinetic theory they use ideal gas model, according to which it is believed that:

1) the intrinsic volume of gas molecules is negligible compared to the volume of the container;

2) there are no interaction forces between gas molecules;

3) collisions of gas molecules with each other and with the walls of the vessel are absolutely elastic.

Let's estimate the distances between molecules in a gas. Under normal conditions (norm: р=1.03·10 5 Pa; t=0ºС) the number of molecules per unit volume: . Then the average volume per molecule:

(m 3).

Average distance between molecules: m. Average diameter of a molecule: d»3·10 -10 m. The intrinsic dimensions of a molecule are small compared to the distance between them (10 times). Consequently, particles (molecules) are so small that they can be likened to material points.

In a gas, molecules are so far apart most of the time that the interaction forces between them are practically zero. It can be considered that the kinetic energy of gas molecules is much greater than the potential energy, therefore the latter can be neglected.

However, in moments of short-term interaction ( collisions) interaction forces can be significant, leading to an exchange of energy and momentum between molecules. Collisions serve as the mechanism by which a macrosystem can transition from one energy state accessible to it under given conditions to another.

The ideal gas model can be used in the study of real gases, since under conditions close to normal (for example, oxygen, hydrogen, nitrogen, carbon dioxide, water vapor, helium), as well as at low pressures and high temperatures, their properties are close to ideal gas.

The state of the body can change when heated, compressed, changed in shape, that is, when any parameters change. There are equilibrium and nonequilibrium states of the system. Equilibrium state is a state in which all system parameters do not change over time (otherwise it is nonequilibrium state), and there are no forces capable of changing the parameters.

The most important parameters of the state of the system are the density of the body (or the inverse value of density - specific volume), pressure and temperature. Density (r) is the mass of a substance per unit volume. Pressure (R– force acting per unit surface area of ​​a body, directed normal to this surface. Difference temperatures (DT) – a measure of the deviation of bodies from the state of thermal equilibrium. There is empirical and absolute temperature. Empirical temperature (t) is a measure of the deviation of bodies from the state of thermal equilibrium with melting ice under pressure of one physical atmosphere. The unit of measurement adopted is 1 degree Celsius(1 o C), which is determined by the condition that melting ice under atmospheric pressure is assigned 0 o C, and boiling water at the same pressure is assigned 100 o C, respectively. The difference between absolute and empirical temperature lies, first of all, in the fact that absolute temperature is measured from the extremely low temperature - absolute zero, which lies below the ice melting temperature by 273.16 o, that is

R= f(V,T).

(6.2.2,b)

Note that any functional relationship that connects thermodynamic parameters like (6.2.2,a) is also called the equation of state. The form of the dependence function between the parameters ((6.2.2,a), (6.2.2,b)) is determined experimentally for each substance. However, so far it has been possible to determine the equation of state only for gases in rarefied states and, in approximate form, for some compressed gases.