An ideal gas. Ideal gas equation.

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1 S t r o n a 1 Autor: Ryszard Świda An ideal gas. Ideal gas equation. To facilitate the application of physical theories, various physicals models are created and used. A physical model is a hypothetical picture of our imagination of a structure, properties and the behavior of matter. To explain thermal phenomena occurring in solid bodies, we have to make certain assumption concerning the structure of these bodies. Solid bodies whose elements may be molecules, atoms or ions have granular structure. Most often these elements constitute configurations called crystals, though there are also solid bodies with random distribution of molecules (amorphous e.g.: fats, ordinary glass). The properties of crystal and amorphous bodies obviously differ. Molecules of a solid body can move in oscillatory and rotation manner (when molecules are polyatomic). Between molecules there are interactions causing potential energy creation, these can be forces of attracting or repelling (depending on the distance). Liquids are also comprised of molecules remaining in constant and random motion. The distances between the liquid molecules are so small that interaction forces do not allow for the complete freedom of movement but their moves are not so limited as in solid bodies. Experiments demonstrate that all volatile bodies have similar properties. Thus, a possibility appears to ignore little differences to focus on common explanation of typical phenomena. Therefore, a physical model has been created which represents all the volatile bodies, the ideal gas model. In this model assumptions of kinetic-molecular theory have been developed and it has been led to form facilitating explanation of phenomena taking place in this state of aggregation. An idealized picture of our imaginations is based on the following assumptions: gas is comprised of a great number of molecules, size of molecules is so small that we can treat them as material points, molecules constantly and randomly move, colliding with each other or with vessel walls, collisions are of elastic character (the principles of conservation of momentum and energy are applied), during the motion the laws of mechanics are applied, forces of molecule interaction exist only at the time of collisions. The simplification introduced in the ideal gas model result in real gases showing some deviations, especially for low temperatures and under a very high pressure. In conditions similar to normal (p n = Pa, T n = 273,15 K) the behaviour of almost all gases is the same as if they were ideal gases.

2 S t r o n a 2 Figure 1 Remember that talking about the number of molecules we mean the number such as or bigger. Thus, it is impossible to follow the behaviour of each molecule. In this model to describe the phenomena we will use the average values: average free road, average speed, average kinetic energy and average number of collisions. We will also apply certain methods used in statistics. Maxwell assumed that the speed distribution when the number of molecules is great and they move in any possible direction is the same as of other random events. The distribution of the velocity has been shown in figure 1. The greatest number of molecules N p has the velocity v p, a little smaller number N has the average value v, and one more again a bit smaller N k so called average square value v śr (coordinates of points A, B, C in the green diagram). In the set of moving molecules there are also molecules with a very slow velocity (even of zero value), but also of very high (practically unlimited). The observations of velocity distributions of the same substance in various temperatures lead to the conclusion that the number of molecules with velocity v p, v and v śr increase when the temperature decreases (similarly points A 1, B 1, C 1 in the blue diagram), whereas velocity values decrease. The ideal gas equation To describe thermo-dynamical phenomena we use so called parameters of gas, these are pressure p, temperature T, volume V and the amount of substance n. What are these parameters and from what have they resulted? We will try to explain that. It is the easiest with volume this is a part

3 S t r o n a 3 of space occupied by gas, limited by vessel s walls, volume equals the capacity of vessel (because gas occupies the whole vessel). The amount of substance is the number of matter expressed in moles creating a given body. Between the amount of substance and the number of molecules there is a following relation: n= N N = N A 6, mol 1 =N 1, mol where: n the amount of substance in moles, N the number of molecules (atoms, molecules, ions), N A Avogadro constant (also called Avogadro number). The pressure from the definition is the thrust (the component of the force acting perpendicularly on the surface) on the unitary surface, therefore, Figure 2 where p is pressure, F thrust, and S is the area size. To explain the dependence of pressure on other values, imagine the following experiment. In a cubical vessel (Figure 3), with the edge a, there are N molecules of an ideal gas which constantly move at an average speed v; directions, values and turns of speed change after collisions. Assume that directions and turns of speed in any direction are equally probable. Thus, we can assume that in each vessel s wall at the same time the same number of molecules hit. If we accept that the distribution of velocity is equal along each of marked directions, it is as if between two parallel walls 1/3 of all molecules move, and at the time t= a v, N/6 number of molecules will hit the wall S. Assume that at the time t there will be so many hits in the wall: By molecules, where N 0 =N x vts 6

4 S t r o n a 4 N x = N V Is the density of gas molecules. Figure 3. At the time of molecule hitting the wall the change of momentum occurs Δ (mv )=F m t because the velocity turn/return changes into the opposite, thus: Δ (mv)=mv ( mv)=2mv Therefore : F m t=2mv so The whole force acting on a wall equals: F m = 2mv t F=N 0 F m =2N 0 mv t F= 2N x vtsmv 6t = N x msv 2 3 While considering the theory of an ideal gas, taking into account a random movement in all possible directions, with various velocity values, instead of velocity square we use the average

5 S t r o n a 5 value of velocity square. The average value of a velocity square is the arithmetic average of the sum of velocity squares of all the molecules velocities. v 2 śr = v 2 1+v v N N After substituting and transforming formulas we will obtain: p= N 2 x mv śr 3 Assume: E kśr = mv 2 śr 2 then: p= 2N x E kśr 3 Or after taking into account density N x we receive: p= 2NE k śr 3V This formula is called the basic formula of kinetic-molecular theory of gas structure and it is applied in relation to an ideal gas as well as real gases in limited scope. In our case we have calculated the pressure acting on one wall, it does not matter because according to Pascal s law the pressure is exerted in all directions in the same way. In fact, molecules collide while moving between the vessel s walls does not change anything, too, because here we deal with elastic collisions during which the rules of conservation of momentum and energy are valid. On the basis of the above consideration it may be stated that pressure is the result of molecules hitting the vessel s walls, and its value depends on density, velocity value, and the mass of molecules. To compare, in Table 1 we give values of: average square velocity, molecular mass, the mass of molecules, average kinetic energy, average velocity, and the number of collisions at the time of 1s of given gases.

S t r o n a 6 Table1* A lot of studies have been conducted which show that molecules of most gases (in temperature of about 300 K) have almost the same average kinetic energy (see data in table 1).

6 S t r o n a 6 Table1* A lot of studies have been conducted which show that molecules of most gases (in temperature of about 300 K) have almost the same average kinetic energy (see data in table 1). Table 2* During further measurements it became obvious that with the change of temperature the average kinetic energy of molecules of all gases changes in direct proportion to temperature measured in absolute temperature scale. Table 2 shows the results of measuring the average square velocity and average kinetic energy of oxygen molecules in dependence of temperature. From the above data it may be concluded that energy and temperature are in direct proportion to each other and the coefficient of proportion equals about C=2, J K. This relation can be expressed with the following formula: or assuming: E kśr T =C E kśr =CT C= 3k 2 The constant k is called Boltzmann constant,

7 S t r o n a 7 k=1, J K E k śr = 3 2 kt As a result of the above relation, absolute zero temperature (0K) would correspond to the standstill of molecules. Using the derived formula for gas pressure, taking into consideration the relation between the average kinetic energy and the temperature, we will receive a very important equation: or p= NkT V pv T =Nk As can be seen from the equation the change of any parameter must cause the change of at least one more parameter. If 1 mole of gas was subjected to change then N = N A = 6, , thus: N A k=8,31 J K =R, R - called gas constant, and the equation will be of the following form: pv T =R If n moles of an ideal gas were subjected to the change, then the equation will be of the following form: or pv T =nr pv=nrt The above equation is called an ideal gas equation. Between the amount of substance, the mass and molecular mass there is the following relation: n= m μ where m is the mass of solid, and µ is molecular mass. Including the above relation in the ideal gas equation we will receive:

8 S t r o n a 8 pv= m μ RT The above equation is called ideal gas law. It may be applied to describe the ideal gas, as well as all real gases, With minor amendments. Problems. 1.What temperature does hydrogen have, whose atoms move at the average velocity of 2700 m/s, and its atomic mass equals 1,67۰10-27 kg? 2. How will the average velocity of gas molecules change if the temperature will be increased twice? 3. The surface of the Sun has the temperature of about 6000K. Why don t hydrogen atoms which are the main component of photosphere leave the Sun? * All the data values given according to: Mieczysław Jeżewski i Józef Kalisz Tablice wielkości fizycznych.

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