The concept of temperature is one of the most important in molecular physics.
Temperature is a physical quantity that characterizes the degree of heating of bodies.
The random chaotic movement of molecules is calledthermal movement.
The kinetic energy of thermal motion increases with increasing temperature. At low temperatures the average kinetic energy of a molecule may be small. In this case, the molecules condense into a liquid or solid; in this case, the average distance between the molecules will be approximately equal to the diameter of the molecule. As the temperature increases, the average kinetic energy of a molecule becomes greater, the molecules fly apart, and a gaseous substance is formed.
The concept of temperature is closely related to the concept thermal equilibrium. Bodies in contact with each other can exchange energy. The energy transferred from one body to another during thermal contact is called amount of heat.
Let's look at an example. If you put heated metal on ice, the ice will begin to melt, and the metal will begin to cool until the temperatures of the bodies become the same. Upon contact between two bodies different temperatures heat exchange occurs, as a result of which the energy of the metal decreases, and the energy of the ice increases.
Energy during heat exchange is always transferred from a body with more high temperature to a body with a lower temperature. Ultimately, a state of the system of bodies occurs in which there will be no heat exchange between the bodies of the system. This condition is called thermal equilibrium.
Thermal equilibrium– This is a state of a system of bodies in thermal contact in which there is no heat transfer from one body to another, and all macroscopic parameters of the bodies remain unchanged.
Temperature– This physical parameter, the same for all bodies in thermal equilibrium. The possibility of introducing the concept of temperature follows from experience and is called the zeroth law of thermodynamics.
Bodies in thermal equilibrium have the same temperatures.
To measure temperatures, the property of a liquid to change volume when heated (and cooled) is most often used.
The device with which temperature is measured is calledthermometer.
To create a thermometer, you must select a thermometric substance (for example, mercury, alcohol) and a thermometric quantity that characterizes the property of the substance (for example, the length of a mercury or alcohol column). IN various designs A variety of thermometers are used physical properties substances (for example, changes in linear dimensions solids or change electrical resistance conductors when heated). Thermometers must be calibrated. To do this, they are brought into thermal contact with bodies whose temperatures are considered given. Most often they use simple natural systems, in which the temperature remains unchanged despite heat exchange with environment is a mixture of ice and water and a mixture of water and steam when boiling at normal atmospheric pressure.
Ordinary liquid thermometer consists of a small glass reservoir to which is attached a glass tube with a narrow internal channel. The reservoir and part of the tube are filled with mercury. The temperature of the medium in which the thermometer is immersed is determined by its position top level mercury in the tube. It was agreed to mark the divisions on the scale as follows. The number 0 is placed in the place of the scale where the level of the liquid column is established when the thermometer is lowered into melting snow (ice), the number 100 is placed in the place where the level of the liquid column is established when the thermometer is immersed in water vapor boiling at normal pressure (10 5 Pa). The distance between these marks is divided into 100 equal parts, called degrees. This method of dividing the scale was introduced by Celsius. Degrees on the Celsius scale are denoted ºC.
By temperature Celsius scale The melting point of ice is assigned a temperature of 0 °C, and the boiling point of water is assigned a temperature of 100 °C. The change in the length of the liquid column in the capillaries of the thermometer by one hundredth of the length between the marks of 0 °C and 100 °C is taken equal to 1 °C.
Widely used in a number of countries (USA) Fahrenheit (T F), in which the freezing temperature of water is taken to be 32 °F and the boiling point of water is 212 °F. Hence,
Mercury thermometers used to measure temperature in the range from -30 ºС to +800 ºС. Along with liquid mercury and alcohol thermometers are used electric And gas thermometers.
Electric thermometer – resistance temperature – it uses the dependence of metal resistance on temperature.
A special place in physics is occupied by gas thermometer , in which the thermometric substance is a rarefied gas (helium, air) in a vessel of constant volume ( V= const), and the thermometric quantity is gas pressure p. Experience shows that gas pressure (at V= const) increases with increasing temperature measured on the Celsius scale.
To calibrate a gas thermometer of constant volume, you can measure pressure at two temperatures (for example, 0 °C and 100 °C), plot points p 0 and p 100 on the graph and then draw a straight line between them. Using the calibration curve thus obtained, temperatures corresponding to other pressure values can be determined.
Gas thermometers are bulky and inconvenient to use. practical application: They are used as a precision standard for calibrating other thermometers.
The readings of thermometers filled with different thermometric bodies usually differ slightly. To precise definition temperature did not depend on the substance filling the thermometer, introduced thermodynamic temperature scale.
To introduce it, let us consider how gas pressure depends on temperature when its mass and volume remain constant.
Thermodynamic scale temperatures Absolute zero.
Let's take a closed vessel with gas and heat it, initially placing it in melting ice. We determine the gas temperature t using a thermometer, and the pressure p using a manometer. As the temperature of the gas increases, its pressure will increase. I found such a dependence French physicist Charles. A graph of p versus t, constructed on the basis of such an experiment, looks like a straight line.
If we continue the graph into the area low pressures, it is possible to determine some “hypothetical” temperature at which the gas pressure would become equal to zero. Experience shows that this temperature is –273.15 °C and does not depend on the properties of the gas. It is impossible to experimentally obtain a gas in a state of zero pressure by cooling, since at very low temperatures all gases turn into liquid or solid states. Pressure ideal gas determined by the impacts of chaotically moving molecules on the walls of the vessel. This means that the decrease in pressure during gas cooling is explained by a decrease average energy forward motion gas molecules E; The gas pressure will be zero when the energy of translational motion of the molecules becomes zero.
The English physicist W. Kelvin (Thomson) put forward the idea that the obtained value of absolute zero corresponds to the cessation of the translational motion of the molecules of all substances. Temperatures below absolute zero cannot exist in nature. This is the limiting temperature at which the pressure of an ideal gas is zero.
The temperature at which the forward motion of molecules should stop is calledabsolute zero ( or zero Kelvin).
Kelvin in 1848 proposed using the point of zero gas pressure to build a new temperature scale –thermodynamic temperature scale(Kelvin scale). The temperature of absolute zero is taken as the starting point for this scale.
In the SI system, the unit of temperature measured on the Kelvin scale is called kelvin and denoted by the letter K.
The size of the Kelvin degree is determined so that it coincides with the Celsius degree, i.e. 1K corresponds to 1ºС.
The temperature measured on the thermodynamic temperature scale is designated T. It is called absolute temperature or thermodynamic temperature.
The Kelvin temperature scale is called absolute temperature scale . It turns out to be most convenient when constructing physical theories.
In addition to the point of zero gas pressure, which is called absolute zero temperature , it is enough to take another fixed reference point. In the Kelvin scale, this point is used temperature triple point water(0.01 °C), in which all three phases - ice, water and steam - are in thermal equilibrium. On the Kelvin scale, the temperature of the triple point is taken to be 273.16 K.
Relationship between absolute temperature and scale temperature Celsius expressed by the formula T = 273.16 +t, where t is the temperature in degrees Celsius.
More often they use the approximate formula T = 273 + t and t = T – 273
Absolute temperature cannot be negative.
Gas temperature is a measure of average kinetic energy molecular movements.
In experiments, Charles found the dependence of p on t. The same relationship will exist between p and T: i.e. there is a directly proportional relationship between p and T.
On the one hand, the gas pressure is directly proportional to its temperature, on the other hand, we already know that the gas pressure is directly proportional to the average kinetic energy of the translational motion of molecules E (p = 2/3*E*n). This means E is directly proportional to T.
The German scientist Boltzmann proposed introducing a proportionality coefficient (3/2)k into the dependence of E on T
E = (3/2)kT
From this formula it follows that the average value of the kinetic energy of the translational motion of molecules does not depend on the nature of the gas, but is determined only by its temperature.
Since E = m*v 2 /2, then m*v 2 /2 = (3/2)kT
where does the root mean square speed of gas molecules come from?
The constant value k is called Boltzmann's constant.
In SI it has the value k = 1.38*10 -23 J/K
If we substitute the value of E into the formula p = 2/3*E*n, we get p = 2/3*(3/2)kT* n, reducing, we get p = n* k*T
The pressure of a gas does not depend on its nature, but is determined only by the concentration of moleculesnand gas temperature T.
The relation p = 2/3*E*n establishes a connection between microscopic (values are determined using calculations) and macroscopic (values can be determined from instrument readings) gas parameters, so it is usually called basic molecular equation - kinetic theory gases.
In this lesson we will analyze a physical quantity that is already familiar to us from the eighth grade course - temperature. We will supplement its definition as a measure of thermal equilibrium and a measure of average kinetic energy. We will describe the disadvantages of some and the advantages of other methods of measuring temperatures, introduce the concept of an absolute temperature scale and, finally, derive the dependence of the kinetic energy of gas molecules and gas pressure on temperature.
There are two reasons for this:
- Various thermometers use various substances as an indicator, therefore thermometers react differently to the same temperature change depending on the properties of a particular substance;
- Arbitrariness in choosing the starting point for the temperature scale.
Therefore, such thermometers are not suitable for any accurate temperature measurements. And since the eighteenth century, more accurate thermometers have been used, which are gas thermometers (see Fig. 2)
Rice. 2. Gas thermometer ()
The reason for this is the fact that gases expand equally when the temperature changes by same values. The following applies to gas thermometers:
That is, to measure temperature, either the change in pressure is recorded at a constant volume, or the volume at a constant pressure.
Gas thermometers often use rarefied hydrogen, which, as we remember, fits the ideal gas model very well.
In addition to the imperfection of household thermometers, there is also the imperfection of many scales that are used in everyday life. In particular, the Celsius scale, as the most familiar to us. As with thermometers, these scales are chosen randomly entry level(for the Celsius scale this is the melting point of ice). Therefore, to work with physical quantities, a different, absolute scale is needed.
This scale was introduced in 1848 by the English physicist William Thompson (Lord Kelvin) (Fig. 3). Knowing that as temperatures increase, the thermal speed of movement of molecules and atoms also increases, it is not difficult to establish that as temperatures decrease, the speed will fall and at a certain temperature will sooner or later become zero, as will the pressure (based on the basic MKT equation). This temperature was chosen as the starting point. It is obvious that the temperature cannot reach a value less than this value, which is why it is called “absolute zero temperature”. For convenience, 1 degree on the Kelvin scale was given in accordance with 1 degree on the Celsius scale.
So, we get the following:
Temperature designation - ;
Unit of measurement - K, "kelvin"
Translation to the Kelvin scale:
Therefore, absolute zero temperature is the temperature
Rice. 3. William Thompson ()
Now, to determine temperature as a measure of the average kinetic energy of molecules, it makes sense to generalize the reasoning that we gave in the definition absolute scale temperatures:
So, as we see, temperature is indeed a measure of the average kinetic energy of translational motion. The specific formulaic relationship was derived by the Austrian physicist Ludwig Boltzmann (Fig. 4):
Here is the so-called Boltzmann coefficient. This is a constant numerically equal to:
As we see, the dimension of this coefficient is , that is, it is a kind of conversion factor from the temperature scale to the energy scale, because we now understand that, in fact, we had to measure temperature in energy units.
Now let's look at how the pressure of an ideal gas depends on temperature. To do this, we write the basic MKT equation in the following form:
and substitute into this formula the expression for the relationship between the average kinetic energy and temperature. We get:
Rice. 4. Ludwig Boltzmann ()
On next lesson We will formulate the equation of state of an ideal gas.
References
- Myakishev G.Ya., Sinyakov A.Z. Molecular physics. Thermodynamics. - M.: Bustard, 2010.
- Gendenshtein L.E., Dick Yu.I. Physics 10th grade. - M.: Ilexa, 2005.
- Kasyanov V.A. Physics 10th grade. - M.: Bustard, 2010.
- Great Encyclopedia of Oil and Gas ().
- youtube.com().
- E-science.ru ().
Homework
- Page 66: No. 478-481. Physics. Problem book. 10-11 grades. Rymkevich A.P. - M.: Bustard, 2013. ()
- How is the Celsius temperature scale determined?
- Indicate the temperature range on the Kelvin scale for your city in summer and winter.
- Air consists mainly of nitrogen and oxygen. The kinetic energy of which gas molecules is greater?
- *How does the expansion of gases differ from the expansion of liquids and solids?
In order to compare equation of state of an ideal gas and the basic equation of molecular kinetic theory, let's write them in the most consistent form.
From these relations it is clear that:
(1.48)
quantity which is called constant Boltzmann- coefficient allowing energy movement molecules(average, of course) express V units temperature, and not just in joules, as until now.
As already mentioned, “explain” in physics means establishing a connection between a new phenomenon, in in this case- thermal, with already studied - mechanical movement. This is the explanation of thermal phenomena. It is precisely for the purpose of finding such an explanation that an entire science has now been developed - statisticalphysics. The word “statistical” means that the objects of study are phenomena in which many particles with random (for each particle) properties are involved. The study of such objects in human populations - peoples, populations - is the subject of statistics.
It is statistical physics that is the basis of chemistry as a science, and not as in a cookbook - “drain this and that, you will get what you need!” Why will it work? The answer is in the properties (statistical properties) of the molecules.
Note that, of course, it is possible to use the found relationships between the energy of molecular motion and the gas temperature in another direction to identify the properties of the molecular motion itself, and the properties of the gas in general. For example, it is clear that molecules inside a gas have energy:
(1.50)
This energy is called - internal.Internal energy there is always! Even when the body is at rest and does not interact with any other bodies, it has internal energy.
If the molecule is not a “round ball”, but is a “dumbbell” (diatomic molecule), then the kinetic energy is the sum of the energy of translational motion (only translational motion has actually been considered so far) and rotational motion ( rice. 1.18 ).
Rice. 1.18. Rotation of a molecule
Arbitrary rotation can be thought of as sequential rotation first around an axis x, and then around the axis z.
The energy reserve of such movement should not differ in any way from the reserve of movement in a straight line. The molecule “does not know” whether it is flying or spinning. Then in all formulas it is necessary to put the number “five” instead of the number “three”.
(1.51)
Gases such as nitrogen, oxygen, air, etc. must be considered using the latest formulas.
In general, if for strict fixation of a molecule in space it is necessary i numbers (they say "i degrees of freedom"), That
(1.52)
As they say, “on the floor kT for each degree of freedom."
1.9. Solute as an ideal gas
Ideal gas ideas find interesting applications in explaining osmotic pressure, arising in solution.
Let there be particles of some other dissolved substance among the solvent molecules. As is known, solute particles tend to occupy the entire available volume. The solute expands in exactly the same way as it expandsgas,to occupy the space provided to it.
Just as a gas exerts pressure on the walls of a container, the solute exerts pressure on the boundary that separates the solution from the pure solvent. This extra pressure called osmotic pressure. This pressure can be observed if the solution is separated from the pure solvent semi-tight partition, through which the solvent easily passes, but the solute does not ( rice. 1.19 ).
Rice. 1.19. The emergence of osmotic pressure in the compartment with the dissolved substance
The solute particles tend to push the septum apart, and if the septum is soft, it bulges. If the partition is rigidly fixed, then the liquid level actually shifts, the level the solution in the compartment with the dissolved substance increases (see. rice. 1.19 ).
Raising the solution level h will continue until the resulting hydrostatic pressureρ gh(ρ is the density of the solution) will not be equal to the osmotic pressure. There is complete similarity between gas molecules and solute molecules. Both are far from each other, and both move chaotically. Of course, between the molecules of the solute there is a solvent, and between the molecules of the gas there is nothing (vacuum), but this is not important. No vacuum was used when deriving the laws! It follows that solute particlesin a weak solution they behave in the same way as ideal gas molecules. In other words, osmotic pressure exerted by a solute,equal to the pressure that the same substance would produce in gaseousstate in the same volume and at the same temperature. Then we get that osmotic pressureπ proportional to the temperature and concentration of the solution(number of particles n per unit volume).
(1.53)
This law is called van't Hoff's law, formula ( 1.53 ) -van't Hoff formula.
The complete similarity of van't Hoff's law with the Clapeyron–Mendeleev equation for an ideal gas is obvious.
Osmotic pressure, of course, does not depend on the type of semipermeable septum or the type of solvent. Any solutions with the same molar concentration exert the same osmotic pressure.
The similarity in the behavior of a solute and an ideal gas is due to the fact that in a dilute solution the particles of the solute practically do not interact with each other, just as the molecules of an ideal gas do not interact.
The magnitude of osmotic pressure is often quite significant. For example, if a liter of solution contains 1 mole of solute, then van't Hoff formula at room temperature we have π ≈ 24 atm.
If a solute decomposes into ions (dissociates) during dissolution, then according to the van’t Hoff formula
π V = NkT(1.54)
it is possible to determine the total number N the resulting particles - ions of both signs and neutral (non-dissociated) particles. And therefore you can find out degree dissociation substances. Ions can be solvated, but this circumstance does not affect the validity of the Van't Hoff formula.
Van't Hoff's formula is often used in chemistry to determination of molecularmasses of proteins and polymers. To do this, to the volume solvent V add m gram of the test substance, measure the pressure π. From the formula
(1.55)
find the molecular mass.
We present the formula for the basic equation of the molecular kinetic theory (MKT) of gases:
(where n = N V is the concentration of particles in the gas, N is the number of particles, V is the volume of gas, 〈 E 〉 is the average kinetic energy of the translational motion of gas molecules, υ k v is the root mean square velocity, m 0 is the mass molecules) relates pressure - a macroparameter that is quite simply measured with such microparameters as the average energy of motion of an individual molecule (or in another expression), the mass of a particle and its speed. But by finding only pressure, it is impossible to establish the kinetic energies of particles separately from concentration. Therefore, to find the full extent of microparameters, you need to know some other physical quantity associated with the kinetic energy of the particles that make up the gas. For this value you can take the thermodynamic temperature.
Gas temperature
To determine the gas temperature you need to remember important property, which reports that under equilibrium conditions, the average kinetic energy of molecules in a mixture of gases is the same for different components of this mixture. From of this property it follows that if 2 gases in different vessels are in thermal equilibrium, then the average kinetic energies of the molecules of these gases are the same. This is the property we will use. In addition, experiments have proven that for any gases (with an unlimited number) that are in a state of thermal equilibrium, the following expression is true:
Taking into account the above, we use (1) and (2) and get:
From equation (3) it follows that the value θ, which we used to denote temperature, is calculated in J, in which kinetic energy is also measured. IN laboratory work The temperature in the measurement system is calculated in Kelvin. Therefore, we introduce a coefficient that will remove this contradiction. It is denoted k, measured in JK and equal to 1.38 10 - 23. This coefficient called Boltzmann constant. Thus:
Definition 1
θ = k T (4) , where T is thermodynamic temperature in kelvins.
The relationship between thermodynamic temperature and the average kinetic energy of thermal motion of gas molecules is expressed by the formula:
E = 3 2 k T (5) .
From equation (5) it is clear that the average kinetic energy of thermal motion of molecules is directly proportional to the gas temperature. Temperature is absolute value. The physical meaning of temperature is that, on the one hand, it is determined by the average kinetic energy per molecule. On the other hand, temperature is a characteristic of the system as a whole. Thus, equation (5) shows the connection between the parameters of the macroworld and the parameters of the microworld.
Definition 2
It is known that temperature is a measure of the average kinetic energy of molecules.
You can set the temperature of the system and then calculate the energy of the molecules.
Under conditions of thermodynamic equilibrium, all components of the system are characterized by the same temperature.
Definition 3
The temperature at which the average kinetic energy of molecules equals 0 and the pressure of an ideal gas equals 0 is called absolute zero temperature. Absolute temperature is never negative.
Example 1
It is necessary to find the average kinetic energy of translational motion of an oxygen molecule if the temperature is T = 290 K. And also find the root mean square speed of a water droplet with a diameter d = 10 - 7 m suspended in the air.
Solution
Let's find the average kinetic energy of motion of an oxygen molecule using the equation connecting energy and temperature:
E = 3 2 k T (1 . 1) .
Since all quantities are specified in the measurement system, let’s carry out the calculations:
E = 3 2 1, 38 10 - 23 10 - 7 = 6 10 - 21 J.
Let's move on to the second part of the task. Let us assume that a droplet suspended in the air is a ball (Figure 1
). This means that the mass of the droplet can be calculated as:
m = ρ · V = ρ · π d 3 6 .
Figure 1
Let's find the mass of a drop of water. According to reference materials, density of water in normal conditions equals ρ = 1000 k g m 3, then:
m = 1000 · 3, 14 6 10 - 7 3 = 5, 2 · 10 - 19 (k g).
The mass of the droplet is too small, therefore, the droplet itself is comparable to a gas molecule, and then the formula for the root mean square velocity of the drop can be used in calculations:
E = m υ k υ 2 2 (1 . 2) ,
where we have already established 〈 E 〉, and from (1. 1) it is clear that the energy does not depend on the type of gas, but depends only on the temperature. This means that we can apply the resulting amount of energy. Let us find the speed from (1.2):
υ k υ = 2 E m = 6 2 E π ρ d 3 = 3 2 k T π ρ d 3 (1 . 3) .
Let's calculate:
υ k υ = 2 6 10 - 21 5, 2 10 - 19 = 0, 15 m s
Answer: The average kinetic energy of translational motion of an oxygen molecule at a given temperature is 6 · 10 - 21 J. The mean square speed of a water droplet at given conditions equals 0.15 m/s.
Example 2
The average energy of translational motion of molecules of an ideal gas is equal to 〈 E 〉, and the gas pressure is p. It is necessary to find the concentration of gas particles.
Solution
The solution to the problem is based on the equation of state of an ideal gas:
p = n k T (2 . 1) .
Let us add to equation (2.1) the equation for the relationship between the average energy of translational motion of molecules and the temperature of the system:
E = 3 2 k T (2 . 2) .
From (2.1) we express the required concentration:
n = p k T 2 . 3.
From (2.2) we express k T:
k T = 2 3 E (2 . 4) .
We substitute (2.4) into (2.3) and get:
Answer: The particle concentration can be found using the formula n = 3 p 2 E.
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The basic equation of the molecular kinetic theory (MKT) of gases:
(where $n=\frac(N)(V)$ is the concentration of particles in the gas, N is the number of particles, V is the volume of gas, $\left\langle E\right\rangle \ $ is the average kinetic energy of translational motion molecules in a gas, $\left\langle v_(kv)\right\rangle $ - root mean square velocity, $m_0$ - molecular mass) connects pressure - a macro parameter that is quite easy to measure with micro parameters - the average energy of motion of an individual molecule or, in another spelling, the mass of the particle and its speed. However, by measuring only pressure, it is impossible to determine the kinetic energies of particles separately from concentration. Consequently, in order for us to be able to fully find microparameters, we need to know some other physical quantity, which is related to the kinetic energy of the particles that make up the gas. This is thermodynamic temperature.
Gas temperature
In order to determine what is gas temperature, it is necessary to recall an important property, which says that at equilibrium the average kinetic energy of molecules in a mixture of gases is the same for different components of this mixture. From this property it follows that if two gases in different vessels are in thermal equilibrium, then the average kinetic energies of the molecules of these gases are the same. We use this property. In addition, experiments have proven that for any gases (the number of gases is not limited) that are in a state of thermal equilibrium, the following relationship holds:
Taking into account the above, we use (1) and (2), we get:
From equation (3) it turns out that the quantity $\theta $, which we introduce as temperature, is measured, like energy, in J. In practice, temperature in the SI system is measured in kelvins. Therefore, we introduce a coefficient that will eliminate this contradiction, its dimension will be $\frac(J)(K)$, the designation k is equal to $1.38\cdot (10)^(-23)$. This coefficient is called Boltzmann's constant. So:
\[\theta =kT\ \left(4\right),\]
where T is the thermodynamic temperature in Kelvin.
And its connection with the average kinetic energy of motion of gas molecules is obvious:
\[\left\langle E\right\rangle =\frac(3)(2)kT\ \left(5\right).\]
Equation (5) shows that the average energy of thermal motion of molecules is directly proportional to the temperature of the gas. The temperature was called absolute. Her physical meaning is that it is determined by the average kinetic energy per molecule. This is on the one hand. On the other hand, temperature is a characteristic of the system as a whole. Thus, equation (5) connects the parameters of the macroworld with the parameters of the microworld. Temperature is said to be a measure of the average kinetic energy of molecules. We can measure the temperature of the system and then calculate the energy of the molecules.
Absolute zero temperatures
In a state of thermodynamic equilibrium, all parts of the system have the same temperature. The temperature at which the average kinetic energy of molecules is zero and the pressure of an ideal gas is zero is called absolute zero temperature. Absolute temperature cannot be negative.
Example 1
Task: Calculate the average kinetic energy of translational motion of an oxygen molecule at a temperature T=290K. The root mean square speed of a water droplet with diameter d=$(10)^(-7)m$ suspended in the air.
You can find the average kinetic energy of motion of an oxygen molecule using an equation connecting it (energy) and temperature:
\[\left\langle E\right\rangle =\frac(3)(2)kT\left(1.1\right).\]
Let's carry out the calculation, since all quantities are given in SI:
\[\left\langle E\right\rangle =\frac(3)(2)\cdot 1.38\cdot (10)^(-23)\cdot (10)^(-7)=6\cdot ( 10)^(-21)\left(J\right).\]
Let's move on to the second part of the problem. A droplet of water that is suspended in the air can be considered a ball (Fig. 1). Therefore, we find the mass of the droplet as $m=\rho \cdot V=\rho \cdot \pi (\frac(d)(6))^3.$
Let's calculate the mass of a water droplet; from reference materials, the density of water under normal conditions is $\rho =1000\frac(kg)(m^3)$:$\ then$
The mass of the droplet is very small, therefore, the droplet itself can be compared to a gas molecule and the formula can be used to calculate the root mean square velocity of the droplet:
\[\left\langle E\right\rangle =\frac(m(\left\langle v_(kv)\right\rangle )^2)(2)\ \left(1.2\right),\]
where $\left\langle E\right\rangle $ we have already calculated, and from (1.1) it is obvious that the energy does not depend on the type of gas, it depends only on the temperature, therefore, we can use the obtained energy value. Let us express the speed from (1.2): $\ \cdot $
\[\left\langle v_(kv)\right\rangle =\sqrt(\frac(2\left\langle E\right\rangle )(m))=\sqrt(\frac(6\cdot 2\left\ langle E\right\rangle )(\pi \rho d^3))=3\sqrt(\frac(2kT)(\pi \rho d^3))\ \left(1.3\right)\]
Let's do the calculation:
\[\left\langle v_(kv)\right\rangle =\sqrt(\frac(2\cdot 6\cdot (10)^(-21))(5.2\cdot (10)^(-19) ))=0.15\ \left(\frac(m)(s)\right)\]
Answer: The average kinetic energy of translational motion of an oxygen molecule at a given temperature is $6\cdot (10)^(-21)\ J$. The root mean square speed of a water droplet under given conditions is 0.15 m/s.
Example 2
Assignment: The average energy of translational motion of molecules of an ideal gas is equal to $\left\langle E\right\rangle .\ $Gas pressure p. Find the concentration of gas particles.
To it we add the equation for the relationship between the average energy of translational motion of molecules and the temperature of the system:
\[\left\langle E\right\rangle =\frac(3)(2)kT\ \left(2.2\right)\]
From (2.1) we express the desired concentration:
From $\left(2.2\right)\ $we express $kT$:
Let's substitute (2.4) into (2.3):
Answer: The concentration of gas particles can be found as $n=\frac(3p)(2\left\langle E\right\rangle )$.
