CRYSTAL PHYSICS
PHYSICAL PROPERTIES OF CRYSTALS
Density
Density is a physical quantity determined for a homogeneous substance by the mass of its unit volume. For an inhomogeneous substance, the density at a certain point is calculated as the limit of the ratio of the mass of the body (m) to its volume (V), when the volume contracts to this point. The average density of a heterogeneous substance is the ratio m/V.
The density of a substance depends on its mass atoms, of which it consists, and on the packing density of atoms and molecules in the substance. The greater the mass of atoms, the greater the density.
But, if we consider the same substance in different states of aggregation, we will see that its density will be different!
A solid is a state of aggregation of a substance, characterized by stability of shape and the nature of the thermal movement of atoms, which perform small vibrations around equilibrium positions. Crystals are characterized by spatial periodicity in the arrangement of equilibrium positions of atoms. In amorphous bodies, atoms vibrate around randomly located points. According to classical concepts, the stable state (with a minimum of potential potential energy) of a solid is crystalline. An amorphous body is in a metastable state and over time should transform into a crystalline state, but the crystallization time is often so long that metastability does not appear at all.
The atoms are tightly bound to each other and very tightly packed. Therefore, a substance in a solid state has the highest density.
The liquid state is one of the aggregate states of matter. The main property of a liquid, which distinguishes it from other states of aggregation, is the ability to change its shape indefinitely under the influence of mechanical stresses, even arbitrarily small, while practically maintaining its volume.
The liquid state is usually considered intermediate between a solid and gas: a gas retains neither volume nor shape, but a solid retains both.
The shape of liquid bodies can be determined entirely or partly by the fact that their surface behaves like an elastic membrane. So, water can collect in drops. But a liquid is capable of flowing even under its stationary surface, and this also means that the form (the internal parts of the liquid body) is not preserved.
The packing density of atoms and molecules is still high, so the density of a substance in a liquid state is not very different from the solid state.
Gas is a state of aggregation of a substance, characterized by very weak bonds between its constituent particles (molecules, atoms or ions), as well as their high mobility. Gas particles move almost freely and chaotically in the intervals between collisions, during which a sharp change in the nature of their movement occurs.
The gaseous state of a substance under conditions where the existence of a stable liquid or solid phase of the same substance is possible is usually called vapor.
Like liquids, gases have fluidity and resist deformation. Unlike liquids, gases do not have a fixed volume and do not form a free surface, but tend to fill the entire available volume (for example, a vessel).
The gaseous state is the most common state of matter in the Universe (interstellar matter, nebulae, stars, planetary atmospheres, etc.). The chemical properties of gases and their mixtures are very diverse - from low-active inert gases to explosive gas mixtures. Gases sometimes include not only systems of atoms and molecules, but also systems of other particles - photons, electrons, Brownian particles, as well as plasma.
Liquid molecules do not have a definite position, but at the same time they do not have complete freedom of movement. There is an attraction between them, strong enough to keep them close.
The molecules have very weak bonds with each other and move far away from each other. The packing density is very low, therefore the substance is in a gaseous state
has low density.
2. Types of density and units of measurement
Density is measured in kg/m³ in the SI system and in g/cm³ in the GHS system, the rest (g/ml, kg/l, 1 t/ M3) – derivatives.
For granular and porous bodies there are:
True density, determined without taking into account voids
Apparent density, calculated as the ratio of the mass of a substance to the entire volume it occupies
3. Formula for finding density
Density is found by the formula:
Therefore, the numerical value of the density of a substance shows the mass of a unit volume of this substance. For example, density cast iron 7 kg/dm3. This means that 1 dm3 of cast iron has a mass of 7 kg. The density of fresh water is 1 kg/l. Therefore, the mass of 1 liter of water is equal to 1 kg.
To calculate the density of gases, you can use the formula:
where M is the molar mass of the gas, Vm is the molar volume (under normal conditions it is equal to 22.4 l/mol).
4. Dependence of density on temperature
As a rule, as the temperature decreases, the density increases, although there are substances whose density behaves differently, for example, water, bronze and cast iron. Thus, the density of water has a maximum value at 4 °C and decreases with both increasing and decreasing temperature.
When the state of aggregation changes, the density of a substance changes abruptly: the density increases during the transition from a gaseous state to a liquid and when the liquid solidifies. True, water is an exception to this rule; its density decreases as it solidifies.
For various natural objects, density varies over a very wide range. The intergalactic medium has the lowest density (ρ ~ 10-33 kg/m³). The density of the interstellar medium is about 10-21 kg/M3. The average density of the Sun is approximately 1.5 times higher than the density of water, equal to 1000 kg/M3, and the average density of the Earth is 5520 kg/M3. Osmium has the highest density among metals (22,500 kg/M3), and the density of neutron stars is of the order of 1017÷1018 kg/M3.
5. Densities of some gases
- Density of gases and vapors (0° C, 101325 Pa), kg/m³
Oxygen 1.429
Ammonia 0,771
Krypton 3,743
Argon 1.784
Xenon 5.851
Hydrogen 0,090
Methane 0,717
Water vapor (100° C) 0.598
Air 1.293
Carbon dioxide 1.977
Helium 0.178
Ethylene 1.260
- Density of some types of wood
Wood density, g/cm³
Balsa 0.15
Siberian fir 0.39
Sequoia evergreen 0.41
Horse chestnut 0.56
Edible chestnut 0.59
Cypress 0.60
Bird cherry 0.61
Hazel 0.63
Walnut 0.64
Birch 0.65
Smooth elm 0.66
Larch 0.66
Field maple 0.67
Teak 0.67
Switenia (Mahogany) 0.70
Sycamore 0.70
Zhoster (buckthorn) 0.71
Lilac 0.80
Hawthorn 0.80
Pecan (cariah) 0.83
Sandalwood 0.90
Boxwood 0.96
Ebony persimmon 1.08
Quebracho 1.21
Gweyakum, or backout 1.28
- Densitymetals(at 20°C) t/M3
Aluminum 2.6889
Tungsten 19.35
Graphite 1.9 - 2.3
Iron 7.874
Gold 19.32
Potassium 0.862
Calcium 1.55
Cobalt 8.90
Lithium 0.534
Magnesium 1.738
Copper 8.96
Sodium 0.971
Nickel 8.91
Tin(white) 7.29
Platinum 21.45
Plutonium 19.25
Lead 11.336
Silver 10.50
Titan 4.505
Cesium 1.873
Zirconium 6.45
- Density of alloys (at 20°C)) t/M3
Bronze 7.5 - 9.1
Wood's Alloy 9.7
Duralumin 2.6 - 2.9
Constantan 8.88
Brass 8.2 - 8.8
Nichrome 8.4
Platinum-iridium 21.62
Steel 7.7 - 7.9
Stainless steel (average) 7.9 - 8.2
grades 08Х18Н10Т, 10Х18Н10Т 7.9
grades 10Х17Н13М2Т, 10Х17Н13М3Т 8
grades 06ХН28МТ, 06ХН28МДТ 7.95
grades 08Х22Н6Т, 12Х21Н5Т 7.6
White cast iron 7.6 - 7.8
Gray cast iron 7.0 - 7.2
Let us place iron and aluminum cylinders of the same volume on the scales (Fig. 122). The balance of the scales has been disrupted. Why?
Rice. 122
In lab work, you measured body weight by comparing the weight of weights to your body weight. When the scales were in equilibrium, these masses were equal. Disequilibrium means that the masses of the bodies are not the same. The mass of the iron cylinder is greater than the mass of the aluminum cylinder. But the volumes of the cylinders are equal. This means that a unit volume (1 cm3 or 1 m3) of iron has a greater mass than aluminum.
The mass of a substance contained in a unit volume is called the density of the substance. To find density, you need to divide the mass of a substance by its volume. Density is denoted by the Greek letter ρ (rho). Then
density = mass/volume
ρ = m/V.
The SI unit of density is 1 kg/m3. The densities of various substances are determined experimentally and are presented in Table 1. Figure 123 shows the masses of substances known to you in a volume V = 1 m 3.
Rice. 123
Density of solids, liquids and gases
(at normal atmospheric pressure)
How do we understand that the density of water is ρ = 1000 kg/m3? The answer to this question follows from the formula. The mass of water in a volume V = 1 m 3 is equal to m = 1000 kg.
From the density formula, the mass of a substance
m = ρV.
Of two bodies of equal volume, the body with the greater density of matter has the greater mass.
Comparing the densities of iron ρ l = 7800 kg/m 3 and aluminum ρ al = 2700 kg/m 3, we understand why in the experiment (see Fig. 122) the mass of an iron cylinder turned out to be greater than the mass of an aluminum cylinder of the same volume.
If the volume of a body is measured in cm 3, then to determine the body mass it is convenient to use the density value ρ, expressed in g/cm 3.
The substance density formula ρ = m/V is used for homogeneous bodies, that is, for bodies consisting of one substance. These are bodies that do not have air cavities or do not contain impurities of other substances. The purity of the substance is judged by the measured density. Is there, for example, any cheap metal added inside a gold bar?
Think and answer
- How would the balance of the scales change (see Fig. 122) if instead of an iron cylinder a wooden cylinder of the same volume were placed on a cup?
- What is density?
- Does the density of a substance depend on its volume? From the masses?
- In what units is density measured?
- How to move from the unit of density g/cm 3 to the unit of density kg/m 3?
Interesting to know!
As a rule, a substance in the solid state has a density greater than in the liquid state. The exception to this rule is ice and water, consisting of H 2 O molecules. The density of ice is ρ = 900 kg/m 3, the density of water? = 1000 kg/m3. The density of ice is less than the density of water, which indicates a less dense packing of molecules (i.e., greater distances between them) in the solid state of the substance (ice) than in the liquid state (water). In the future, you will encounter other very interesting anomalies (abnormalities) in the properties of water.
The average density of the Earth is approximately 5.5 g/cm 3 . This and other facts known to science allowed us to draw some conclusions about the structure of the Earth. The average thickness of the earth's crust is about 33 km. The earth's crust is composed primarily of soil and rocks. The average density of the earth's crust is 2.7 g/cm 3, and the density of the rocks lying directly under the earth's crust is 3.3 g/cm 3. But both of these values are less than 5.5 g/cm 3, i.e. less than the average density of the Earth. It follows that the density of matter located in the depths of the globe is greater than the average density of the Earth. Scientists suggest that in the center of the Earth the density of the substance reaches 11.5 g/cm 3, that is, it approaches the density of lead.
The average density of human body tissue is 1036 kg/m3, the density of blood (at t = 20°C) is 1050 kg/m3.
Balsa wood has a low wood density (2 times less than cork). Rafts and lifebelts are made from it. In Cuba, the Eshinomena prickly hair tree grows, the wood of which has a density 25 times less than the density of water, i.e. ρ = 0.04 g/cm 3 . The snake tree has a very high wood density. A tree sinks in water like a stone.
Do it yourself at home
Measure the density of the soap. To do this, use a rectangular shaped bar of soap. Compare the density you measured with the values obtained by your classmates. Are the resulting density values equal? Why?
Interesting to know
Already during the life of the famous ancient Greek scientist Archimedes (Fig. 124), legends were formed about him, the reason for which was his inventions that amazed his contemporaries. One of the legends says that the Syracusan king Heron II asked the thinker to determine whether his crown was made of pure gold or whether the jeweler mixed a significant amount of silver into it. Of course, the crown had to remain intact. It was not difficult for Archimedes to determine the mass of the crown. Much more difficult was to accurately measure the volume of the crown in order to calculate the density of the metal from which it was cast and determine whether it was pure gold. The difficulty was that it was the wrong shape!
Rice. 124
One day, Archimedes, absorbed in thoughts about the crown, was taking a bath, where he came up with a brilliant idea. The volume of the crown can be determined by measuring the volume of water displaced by it (you are familiar with this method of measuring the volume of an irregularly shaped body). Having determined the volume of the crown and its mass, Archimedes calculated the density of the substance from which the jeweler made the crown.
As the legend goes, the density of the crown’s substance turned out to be less than the density of pure gold, and the dishonest jeweler was caught in deception.
Exercises
- The density of copper is ρ m = 8.9 g/cm 3, and the density of aluminum is ρ al = 2700 kg/m 3. Which substance is more dense and by how many times?
- Determine the mass of a concrete slab whose volume is V = 3.0 m 3.
- What substance is a ball with volume V = 10 cm 3 made of if its mass m = 71 g?
- Determine the mass of window glass whose length a = 1.5 m, height b = 80 cm and thickness c = 5.0 mm.
- Total mass N = 7 identical sheets of roofing iron m = 490 kg. The size of each sheet is 1 x 1.5 m. Determine the thickness of the sheet.
- Steel and aluminum cylinders have the same cross-sectional area and mass. Which cylinder has the greater height and by how much?
Everything around us consists of different substances. Ships and bathhouses are built from wood, irons and cots are made from iron, tires on wheels and erasers on pencils are made from rubber. And different objects have different weights - any of us can easily carry a juicy ripe melon from the market, but we will have to sweat over a weight of the same size.
Everyone remembers the famous joke: “Which is heavier? A kilogram of nails or a kilogram of fluff? We will no longer fall for this childish trick, we know that the weight of both will be the same, but the volume will be significantly different. So why is this happening? Why do different bodies and substances have different weights with the same size? Or vice versa, the same weight with different sizes? Obviously, there is some characteristic due to which substances are so different from each other. In physics, this characteristic is called the density of matter and is taught in the seventh grade.
Density of a substance: definition and formula
The definition of the density of a substance is as follows: density shows what the mass of a substance is in a unit of volume, for example, in one cubic meter. So, the density of water is 1000 kg/m3, and ice is 900 kg/m3, which is why ice is lighter and is on top of reservoirs in winter. That is, what does the density of matter show us in this case? An ice density of 900 kg/m3 means that an ice cube with sides of 1 meter weighs 900 kg. And the formula for determining the density of a substance is as follows: density = mass/volume. The quantities included in this expression are designated as follows: mass - m, volume of the body - V, and density is designated by the letter ρ (Greek letter “rho”). And the formula can be written as follows:
How to find the density of a substance
How to find or calculate the density of a substance? To do this you need to know body volume and body weight. That is, we measure the substance, weigh it, and then simply substitute the obtained data into the formula and find the value we need. And how the density of a substance is measured is clear from the formula. It is measured in kilograms per cubic meter. Sometimes they also use a value such as grams per cubic centimeter. Converting one value to another is very simple. 1 g = 0.001 kg, and 1 cm3 = 0.000001 m3. Accordingly, 1 g/(cm)^3 =1000kg/m^3. It should also be remembered that the density of a substance is different in different states of aggregation. That is, in solid, liquid or gaseous form. The density of solids is most often higher than the density of liquids and much higher than the density of gases. Perhaps a very useful exception for us is water, which, as we have already considered, weighs less in the solid state than in the liquid state. It is because of this strange feature of water that life is possible on Earth. Life on our planet, as we know, originated from the oceans. And if water behaved like all other substances, then the water in the seas and oceans would freeze through, the ice, being heavier than water, would sink to the bottom and lie there without melting. And only at the equator, in a small column of water, would life exist in the form of several species of bacteria. So we can say thank you to the water for our existence.
Instructions
So, everyone has long been unaware that the density of a substance, be it a liquid or a solid aggregate, can be calculated as mass divided by volume. That is, in order to experimentally determine the density of ordinary liquid water, you need to: 1) Take a measuring cylinder and weigh it.
2) Pour water into it and record the volume it occupies.
3) Weigh the cylinder with water.
4) Calculate the mass difference, obtaining the mass of water.
5) Calculate the density using the known formula
However, we noticed that the density values differ at different temperatures. But the most amazing thing is the law by which the change occurs. Scientists around the world are still puzzling over this phenomenon. No one can solve the mystery and answer the question: “Why is the density value during heating from 0 to 3.98, and after 3.98?” A couple of years ago, Japanese physicist Masakazu Matsumoto proposed a model for the structure of water molecules. According to this theory, certain polygonal microformations are formed in water - vitrites, which in turn prevail over the phenomenon of elongation of hydrogen bonds and compress water molecules. However, this theory has not yet been confirmed experimentally. A graph of density versus temperature is shown below. To use it you need: 1) Find the temperature value you need on the corresponding axis.
2) Lower the perpendicular to the graph. Mark the point of intersection of the line and the function.
3) From the resulting point, draw a line parallel to the temperature axis to the density axis. The intersection point is the desired value. Example: Let the water temperature be 4 degrees, then the density, after construction, turns out to be equal to 1 g/cm^3. Both of these values are approximate.
To determine a more accurate density value, you need to use the table. If there is no data there for the temperature value you need, then: 1) Find the values between which the desired value is located. For a better understanding, let's look at an example. Let the density of water at a temperature of 65 degrees be required. It is between 60 and 70.
2) Draw a coordinate plane. Specify the x-axis as temperature and the y-axis as density. Mark the points you know on the graph (A and B). Connect them with a straight line.
3) Lower the perpendicular from the temperature value you need to the segment obtained above, mark it as point C.
4) Mark points D, E, F as shown on the graph.
5) Now it is clearly visible that triangles ADB and AFC are similar. Then the following relation is valid:
AD/AF=DB/EF, therefore:
(0.98318-0.97771)/(0.98318-x)=(70-60)/(65-60);
0.00547/(0.98318-x)=2
1.96636-2x=0.00547
x=0.980445
Accordingly, the density of water at 65 degrees is 0.980445 g/cm^3
This method of finding a value is called the interpolation method.
Definition
Density of matter (density of body matter) is a scalar physical quantity that is equal to the ratio of the mass (dm) of a small element of a body to its unit volume (dV). Most often, the density of a substance is denoted by a Greek letter. So:
Types of density of matter
Using expression (1) to determine density, we speak about the density of the body at a point.
The density of a body depends on the material of the body and its thermodynamic state.
where m is body mass, V is body volume.
If the body is inhomogeneous, then sometimes they use the concept of average density, which is calculated as:
where m is body mass, V is body volume. In technology, for inhomogeneous (for example, granular) bodies, the concept of bulk density is used. Bulk density is calculated in the same way as (3). The volume is determined by including spaces in bulk and loose materials (such as sand, gravel, grain, etc.).
When considering gases under normal conditions, the formula is used to calculate the density:
where is the molar mass of the gas, is the molar volume of the gas, which under normal conditions is 22.4 l/mol.
Units for measuring the density of matter
In accordance with the definition, we can write that the units of measurement of density in the SI system are: = kg/m 3
in GHS: =g/(cm) 3
In this case: 1 kg/m 3 = (10) -3 g/(cm) 3.
Examples of problem solving
Example
Exercise. What is the density of water if the volume occupied by one molecule of H 2 O is approximately equal to m 3? Consider that the molecules in water are tightly packed.
where m 0 is the mass of a water molecule. Let's find m 0 using the known relation:
where N=1 is the number of molecules (in our case one molecule), m is the mass of the number of molecules under consideration (in our case m=m 0), N A =6.02 10 23 mol -1 – Avogadro’s constant, =18 10 - 3 kg/mol (since the relative molecular mass of water is M r =18). Therefore, using expression (2) to find the mass of one molecule we have:
Substitute m 0 into expression (1), we get:
Let's calculate the required value:
kg/m 3
Answer. The density of water is 10 3 kg/m 3.
Example
Exercise. What is the density of cesium chloride (CsCl) crystals if the crystals have a cubic crystal lattice (Fig. 1) at the vertices of which there are chlorine ions (Cl -), and in the center there is a cesium ion (Cs +). Consider the edge of the crystal lattice to be d=0.41 nm.
Solution. As a basis for solving the problem, we take the following expression:
where m is the mass of the substance (in our case, this is the mass of one molecule - Avogadro’s constant, 
kg/mol molar mass of Cesium chloride (since the relative molecular mass of cesium chloride is equal to ). Expression (2.1) for one molecule will take the form.
