1. (2.5.1) So, the number of moles, n, contained in a sample of any substance is equal N n, (2.5.2)

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1 Lecture.5. Ideal gas law We have already discussed general rinciles of classical therodynaics. Classical therodynaics is a acroscoic science which describes hysical systes by eans of acroscoic variables, such as ressure, volue, density and teerature. hese quantities are obtained by averaging over large nuber of olecules fro which the syste consists. Introducing these variables, we never discussed icroscoic structure of substances and the relations between this structure and aforeentioned variables. oday we are going to consider one articular exale of therodynaic syste, which includes the so-called ideal gas. We shall also consider this gas not just fro the standoint of classical therodynaics, but also using icroscoic aroach, known as the kinetic theory of gases. We shall find interconnections between the two aroaches. It is obvious fro the beginning that acroscoic ressure of gas exists due to collisions between the gas olecules and the walls of the container. eerature of gas is for sure related to kinetic energy of its olecules. When talking about olecules, it is efficient to introduce a secial unit to easure the aount of substance. Of course, we could do so in ters of nubers of olecules. But the nuber of olecules filling the volue of several cubic eters is so huge that it will be very inconvenient to oerate with such large nubers. Instead, we shall use a secial unit, ole, to easure the aount of substance. Mole is also one of the fundaental units of SI. It is defined as the nuber of atos in a g sale of carbon-. his nuber of atos can be easured exerientally and it is known as the Avogadro's nuber which is N A ol. (.5.) So, the nuber of oles, n, contained in a sale of any substance is equal N n, (.5.) N A where N is the nuber of olecules in the sale. he aount of substance can be calculated if one knows the ass of one ole for the substance (the olar ass M) or the ass of one olecule, so

2 Msa Msa n, (.5.3) M N where M sa is ass of the sale. A Now we are ready to discuss the ideal gas. We can see even fro its nae, that this substance does not actually exist in nature, it is an idealization. However, at certain conditions any gas can be considered as alost ideal. his idealization assues that there are no any interolecular forces acting between the olecules of this gas. In other words, these olecules are neither attracted, not reelled by each other. he otential energy of their interaction is equal to zero. he only tye of energy which they can osses is the kinetic energy of their otion, and it changes only by eans of the rear collisions between the olecules and the walls of the container. his idealization is alost valid for any gas as long as it has low density. In such a case the average distance between the olecules is so large, that we can ignore otential energy of their interaction coared to their kinetic energy. It is exeriental result for gases at low density, as well as theoretical result for ideal gas, that nr. (.5.4) You will actually try to verify that stateent in the lab next tie. his equation is known as the ideal gas law. In that equation, is the absolute ressure of gas, is its volue, is teerature (in Kelvin scale), n is the aount of substance (in oles) and R is the universal gas constant, which is R 83. J ol K. (.5.5) his equation can also be reresented in a different for as where Nk, (.5.6) R J k (.5.7) N K A is the Boltzann constant. Let us calculate work, which can be erfored by ideal gas in the container under the iston at different external conditions. a) We shall first consider isotheral exansion, when gas in the container exands fro original volue to the final volue and at the sae tie it is ket

3 at constant teerature. his can be arranged, if the exansion rocess is quite slow, so this gas has enough tie to reach theral equilibriu with its environent all the tie during this rocess. Pressure of this ideal gas can then be found fro equation.6.4 as nr, In the case of the constant teerature and the constant aount of substance, this deendence will look like hyerbola in the (, ) coordinate lane. his gives, (.5.8) which is known as the Boyle s Law. o find work done by the gas, we need to calculate the integral z z z nr W d d nr d F nrln nr ln H G. (.5.9) Since this gas exands, so W 0 and gas erfors ositive work. If this gas is coressed, which eans ositive work was erfored on the gas, then and W 0 (gas itself erfors the negative work). b) In the case if we have constant volue-rocess, which eans the work W 0, not only for ideal but for any gas. he ideal gas law gives nr const, or, (.5.0) which is know as the Gay-Lussac law. c) In the case if we have constant ressure rocess then the work which is done by this gas is going to be z z W d d not just for ideal but for any gas. hen b g, (.5.) I KJ

4 nr const, or, (.5.) known as the Charles s law. Exale.5. A balloon contains.0 liters of nitrogen gas at a teerature of 77 K and a ressure of 0 kpa. If the teerature of the gas is allowed to increase to 3 deg C and the ressure reains constant, what volue will the gas occuy? Let us consider this nitrogen as an ideal gas. In that case the ideal gas law gives nr. One can rewrite this equation as nr. he right hand side of the equation stays constant for the syste considered here, which eans that the left hand side is also conserved and K has. 0L L. 77K or o. aking into account that 3 C 96K, one. Kinetic heory of Gases So far we have interreted ideal gas law fro the acroscoic stand-oint and essentially took it as exeriental result. Now we have to see what the icroscoic significance of this law is. Our goal is to relate acroscoic characteristics of gas, such as its ressure or teerature to icroscoic ones, such as velocity of gas olecules and their energy. Let us consider the gas in the aount of n oles which is laced in the cubic box of volue. his gas exerts ressure on the walls of the box. he gas and the box are ket at constant teerature. We shall assue that this gas is an ideal gas. So, the tyical gas olecule of ass oves with velocity v and then collides with the wall of the box. We will treat this collision as absolutely elastic collision. So the olecule will change the direction of the coonent of its velocity erendicular to the wall to the oosite direction. Let us call this direction to be the x-direction. his axis will be directed towards the wall, so the change of the olecule's oentu will be

5 x vx vx vx and the change of the wall's oentu, due to this collision, is v x. he next collision will occur after the tie interval t for which this olecule travels to the oosite wall reflects fro it and coes back (we ignore collisions between olecules). his tie is t L v x, where L is the distance between the walls. So, this olecule will deliver oentu to the wall at average rate of t x v L v x x v L x. According to the Newton's second law this rate equals to the average force acting on the wall fro this olecule. o find the total force acting on this wall we have to erfor suation over all the olecules, so finally we have ressure where N vxi L Fx Fx i 3 A L L L N N i v xi nn A is the nuber of olecules in the box. his ressure will be, L N v nn A ( x ) avg ( v 3 3 x ) avg, L v x avg where ( ) is the average value for the square of the olecule s velocity in the x direction taken over all the olecules in the box. Since there are any olecules in the box and they are oving in rando directions, we can say that v x avg v avg the average 3 square of the velocity coonent in one direction is just /3 of the average square of the olecule's seed. So the ressure will becoe nm( v ) 3 avg. One can introduce the root-ean-square seed, which is v rs ( v ) avg, so nmv 3 rs. aking into account the ideal gas law we finally have the icroscoic quantity v rs related to acroscoic teerature as

6 v rs 3R k M 3. (.5.3) For ost of the gases, this velocity at usual roo teerature has a very high value of several hundreds or even thousands of eters er second. elocity of sound in gas cannot be higher than velocity of its olecules. Exerient shows that it is in fact slightly saller than that. elocities of olecules are related to kinetic energy of translational otion of these olecules, so that the average kinetic energy of one olecule will be K avg F H G I K J d i. (.5.4) v v k v rs 3 3 k avg avg At given teerature, all ideal gas olecules, have the sae average translational kinetic energy of 3k. Of course, even though we have already calculated the root-ean square seed of the olecules, it does not ean that all the olecules are oving with this seed. Soe of the are uch faster, while others are uch slower. he faster the olecule is above the average, the fewer nubers of such fast olecules can exist. he sae is true about slow olecules. here are no olecules in gas which are not oving and only a few are oving with sufficiently law seeds. In 85 the Scottish hysicist Jaes Clerk Maxwell derived the distribution of ideal gas olecules over their seeds. His result is known as the Maxwell's seed distribution law, which is 3 b g F H G I K J P v 4 k In this equation P v v e v k. (.5.5) b g is the robability density distribution function. So, the fraction of the olecules, having seeds between v and v +dv, is P(v)dv. he robability for the olecule to have any seed is equal to. So z Pbvgdv =. (.5.6) 0 his condition can be verified by integrating equation.5.5. his distribution allows us to calculate the average seed of gas olecules by couting v z avg b g 0 vp v dv 8k. (.5.7)

7 Exercise: Calculate this integral and rove equation.5.7. Hint: Consider new variable substitution and integration by arts. We can also find the average square for the seed of gas olecules 3k dv i z v Pbvgdv. (.5.8) avg 0 his gives us again the equation.5.3 for the root-ean-square seed. Finally the ost robable seed of the olecules is the one which corresonds to the axiu on the P(v)-curve. his axiu can be found fro the condition dp dv 0. his gives k v. (.5.9) Exercise: Calculate this derivative and rove the equation.5.9. We have been talking about seed of olecules for a long tie, because this seed is related to their kinetic energy. his we have already seen in the equation.5.4. In the case of ideal gas, the kinetic energy is the only tye of energy the gas olecules can have. his eans we can relate our results to the total internal energy of ideal gas. In the case of onatoic gas, which consists of individual atos rather than olecules, such as He, Ne, Ar, the atos can be considered as article-like objects without size. So, they cannot rotate and total internal energy of such gas is just the su of translational kinetic energies of its atos. We have already roved that the average translational kinetic energy of one olecule (ato in this case) is 3k. So, for the gas consisting of N olecules, one has E Nk nn Ak nr. (.5.0) hus, the internal energy of ideal gas is function of the gas teerature only. It does not deend on any other variable. In reality olecules of gas erfor very colicated otion. hey are not oving along the straight lines. Instead they are exeriencing very often collisions with each other, changing their directions and seeds. One useful araeter to describe this rando otion is the ean free ath,, of olecules between their collisions. Of course, this quantity becoes saller with increase of gas density N. It also deends on diaeter of gas olecules, d (assuing that olecules are sheres). Since the effective cross

8 section of a olecule is roortional to the diaeter squared, this eans that should decrease with increase of d. Collisions occur when the distance between the centers of two olecules becoes less than d. So, if a olecule oves for soe tie t without exeriencing collision with other olecules, it oves through the cylinder, which has the radius d and the length v t, where v is the seed of the olecule. hus the volue of that cylinder isd v t. he length of the tie interval between the collisions deends on the density of gas. If there is at least one olecule in this cylinder a collision will occur, so d v t N and t. hen the length of the cylinder is going to be d v N v vt d v N d N. In this aroxiation we have not taken into account the fact that not only this olecule is oving but all other olecules are oving too. If the roer averaging is erfored the extra factor arises and the forula for the ean free ath becoes d N. (.5.)

3 Thermodynamics and Statistical mechanics

3 Thermodynamics and Statistical mechanics Therodynaics and Statistical echanics. Syste and environent The syste is soe ortion of atter that we searate using real walls or only in our ine, fro the other art of the universe. Everything outside the