3 Thermodynamics and Statistical mechanics

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1 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 syste that has a direct effect on its behaviour we call the environent. For exale: the gas in a container can be the syste and the oveable iston and the Bunsen burner can be the environent. We ust choose suitable and observable quantities to describe the behaviour of the syste. The bulk roerties of the syste as acroscoic are called therodynaic variables. There are two classes of these variables The so called extensive state variables are additive quantities as: volue, ass, E energy The ass of a cobined or united syste is the su of the individual asses they are characteristic for a ortion of the syste. The so called intensive state variables are equalizing quantities as: ressure, T teerature The intensive state variables can be ordered to any oint of a syste. Basic concets of therodynaics.. Theral equilibriu When the distribution of the intensive araeters inside a syste is hoogeneous, then, the syste is in theral equilibriu. When all the intensive araeters of two systes which are in equilibriu are the sae; then these systes are in theral equilibriu with each other. The zeroth law of therodynaics states: If and B are each in theral equilibriu with a third body C, then and B are in theral equilibriu with each other... Concet of internal energy The internal energy of a any-article syste include the kinetic energy of the so called disordered olecular otion and the otential energy of the olecular interactions (an der Waals) but doesn t contain the otion of atter in bulk aears to be ordered and the otential energy belongs to the interaction with external fields. The next question arises, how can we change the internal energy of a any-article syste? B. Palásthy

2 .. Work Consider a gas inside a cylinder, one wall of which is a ovable iston. Denote the ressure of the gas by, it is the force er unit area: F = dx If one wall of the container is oveable the force exerted by the gas ay roduce a dislaceent dx of the wall. Therefore the eleentary work done by the gas: dw ' = Fdx = dx = d, where d = dx is the change in volue of the gas. Since the work done by the gas (syste) and the work done by the environent corresond to the sae dislaceents and the two forces equal and oosite, the two works are equal in agnitude but have oosite signs. The eleentary work done by the environent on the syste is: δ W = d The eleentary work done by syste: δ W ' = d δw = δw ' If the volue changes fro to the finite work done by environent is: W = d To coute the integral we ust know the relation between and (that is the equation of state). If this relation is known we can draw the so called - diagra. W The geoetrical eaning of the integral is the area under the curve. So the work done by the syste in going fro to : W ' = d This figure indicates several rocesses between all of which take a syste fro state to state. Since the area under each curve is different, the work done in each rocess is also different. B. Palásthy

3 We can state, that the work deends on the rocess. It is often called rocess characteristic, or rocess variable. If the gas exands d >, and δ W ' = d >, but δ W = d <. If the environent coress the gas d <, and δ W ' = d <, but δ W = d >. If work is done on the syste we can say that its internal energy changes. So this is the so called ordered way to change the internal energy of a syste...4 Heat Consider a gas inside a rigid container. If we bring a hot body, such as a flae, close to the container we observe an increase in teerature and ressure of the gas. This suggests that the energy of the olecules has also increased. This tye of energy transfer is called heat and denoted by Q. This energy transfer is due to the large nuber of energy exchanged between the olecules of the syste and the olecules of the surrounding. The heat is considered Q > ositive if the syste gains energy (so it is absorbed) the heat is Q < negative if the syste loses energy (so it is given off by the syste). If there is no net exchange of energy between two systes, we say that they are in theral equilibriu. In such case there is no net heat transfer.. Conservation of energy, The first low of therodynaics The conservation of the energy for a any-article syste states that the change of internal energy of a syste is equal to the heat absorbed lus the external work done on the syste. Δ E = Q+ W The stateent above is called the first law of therodynaics. It is iortant to recognize that the internal energy of a syste deends only on the state of the syste because it is deterined by the energy distribution of the olecules in the state considered. We can say that the internal energy is a state variable. The change of the internal energy is indeendent of the rocess taken. B. Palásthy

4 Δ E is the sae for both rocesses shown on the figure. Consider the first law of therodynaics: Q= ΔE W The heat Q ust also deend on the ath since W deends but Δ E doesn t deend. The first law for a finite rocess: and for an eleentary sall rocess: or Δ E = Q+ W, de = δ Q + δw de = δ Q d..4 Teerature n iortant concet related to the sensations of hot and cold is that of teerature. When we feel a hot body we say it has high teerature. When we feel a cold body we say its teerature is low. We learned to define and easure the teerature of a body long before we understand its hysical eaning. The instruent used to ake the easureent is called theroeter. Teerature is easured by observing the changes in soe hysical roerty (for exale: the length of a liquid colun in a caillary tube). In the Celsius scale the unit of teerature is a degree, denoted by o C. With the value of o C assigned to the teerature of elting ice, and o C to the teerature of boiling water both at standard atosheric ressure..4. The ideal Gas Teerature Soe roerties of a gas are so sensitive to teerature changes that they ay be used to easure the teerature of the gas. Let us consider a ass of gas that at a certain teerature occuies a volue. The ressure of the gas designated as. Boyle recognised that if the teerature does not vary the roduct of the ressure ties volue reains constant: = constant This result is known as Boyle s law. rocess during which the teerature does not vary is called an isotheral transforation. It is also an exeriental result that, for a given ass of gas the constant deends on the teerature. Therefore, the value of the roduct for a given ass of gas in an indicator of the teerature of the gas. We shall define the gas teerature as a quantity roortional to the roduct, so we ay write: B. Palásthy 4

5 = CT When the gas is at the freezing oint of water the ressure is and volue is, at the boiling oint, and T are the corresonding teerature. = CT, and = CT Next we decide that between T and T we easure units: = C( T T) = C Taking the ratio of the two equations: T =, we can obtain: T = It has been observed that the value of T is: T = 7,5 K. The unit of the gas teerature scale is called Kelvin scale and designated by K. On this scale the freezing oint of water is T = 7,5 K, and the boiling oint is T = 7,5 K. The general rule is: o K = C+ 7,5 If we have n oles of a gas we ay write in the for: = nrt Where R is a new constant, essentially the sae for all gases and is called the gas constant, its exerientally deterined value is: R = 8, J. olk If the ass of the gas is and the olar ass is then: n =, and the state equation is: = RT, = nrt The equation relates the ressure, volue and the teerature is called the equation of state of the gas. Gases follow this equation only at high teeratures and low densities. gas that follows this equation at all teeratures and densities is called an ideal gas, and T is called absolute teerature. If N is the nuber of olecules of the gas, and N is vogadro s nuber, the nuber of oles of the gas is: N n =, and N R = N T N Introducing the Boltzann s constant as: B. Palásthy 5

6 The state equation can be written: k R N = =,8, = NkT J K.5 Ideal Gas. icroscoic Descrition Fro the icroscoic oint of view we define an ideal gas by aking the following assutions:. gas consists of articles called olecules.. The olecules are in rando otion and obey Newton s laws of otion. They ove in all directions and with various seeds.. The total nuber of olecules is large. 4. The volue of the olecules is a negligibly sall fraction of the volue occuied by the gas. 5. No areciable forces act on the olecules excet during a collision. (Between two collisions olecules ove uniforly.) 6. Collisions are erfectly elastic and are of negligible duration. 7. Consider now ono-atoic gas..5. Kinetic calculation of ressure Let us now calculate the ressure of an ideal gas fro kinetic theory. Consider ideal gas in a container. Deterine the force exerted by the olecules collide with the wall. v v x v Let be the ass of a olecule and v its seed. Denote the x coonent of velocity which is erendicular to the wall by v x. fter the elastic collision with the wall, the change of the linear oentu: Δ x = v x. If n is the nuber of olecules in unit volue: N n =. During a given tie Δ t only those olecules can reach the wall and collide, whose distance fro the wall is less than vxδ t. During Δ t tie nvxδ t olecules collide. The total change of linear oentu is: ΔxnvxΔ t. The force exerted by the gas olecules, when they collide with the wall: B. Palásthy 6

7 Δ Δnv x x Δt F = = = vxnvx. Δt Δt The ressure is the force over the area: F = = vn x. It is better to suose that only the half of olecules ove in one direction, and the other half away fro the wall, so we can reduce by : = vn x. ctually, since the olecules ove with different velocities we ost use the average value = vn x. We ay assue that if the gas hoogeneous the average olecular velocity is the sae in any direction: vx = vy = vz, and v = v + v + v. v x y z = v x. N v ly that n =, and v x = : v N = = v N, = v N. Introduce the average kinetic energy ε of a olecule, so: Nε =. In case of ideal gas the internal energy is kinetic energy of the disordered olecular otion, and the an der Waals otential energy is negligible. So Nε = E is just the total internal energy of the ideal gas: E = Coaring this result with the ideal gas equation: = NkT, E = NkT. The average kinetic energy, and so the internal energy of a gas directly roortional to the absolute teerature: = ε N, = NkT, and so: ε = kt B. Palásthy 7

8 ε = kt. The average kinetic energy of a gas olecule is related to the absolute teerature. Since by definition an ideal gas has no otential energy the total internal energy of an ideal gas of N articles is then: E = Nε = NkT Consider again the average kinetic energy of a olecule: ε = kt, and ε = v kt = v, kt v = So the average value of the square of the velocity is roortional to the absolute teerature T. The square root of v is called root-ean-square seed v rs : kt RT vrs = v = = here we used that: N NkT = RT RT =, kt RT =. s we calculated the internal energy of the gas we suosed that the olecules have only rando translational otion and kinetic energy. It is true for a ono-atoic gas. In case of diatoic of olyatoic gas the olecules can rotate beside the translation The nuber of velocity coonents needed to describe the otion of a olecule coletely is called the nuber of degrees of freedo denoted by f: for ono-atoic gas f = for diatoic gas f = 5 for olyatoic gas f = 6 For a diatoic olecule there are two ossible axes of rotation erendicular to each other. Using the concet of freedo energy of the ideal gas: f f E = = NkT The average energy of a olecule: f ε = kt The rincile of equiartition of energy states, that for any degree of freedo for any olecule the sae average energy is associated that is: B. Palásthy 8

9 ε = kt The finite change of the internal energy of the ideal gas with f degree of freedo: f f Δ E = NkΔ T = RΔ T.6 The alication of the First Law of Therodynaics for secial transforations of ideal gas.6. Isochoric transforation In case of isochoric transforation the volue is ket constant: = constant. = constant The connection between ressure and teerature: = RT R = = constant, T =. T T ly the first law of therodynaics: Δ E = Q + W s the volue is constant there is no work done by the environent, W =, therefore: f Q = Δ E = RΔ T Introduce the next exression which is called secific heat at constant volue: f R c = This quantity is different for different gases. The unit is J [ c ] =. kgk The heat can be written as: Q = c Δ T The roduct of the olar ass and the secific heat is called olar secific heat: B. Palásthy 9

10 and its unit is: Using the olar secific heat: f C = c = R, [ C ] Q J = olk = C nδ T..6. Constant-ressure rocess We enclose the gas in a cylinder with a iston that oves so as to aintain constant ressure. = constant = constant The ideal gas state equation: = RT R = = constant T =, T T ly the first law of therodynaics: Δ E = Q + W The change of the internal energy: f Δ E = RΔ T. The work done by the environent in case of constant ressure rocess: ( ) W = d = d = ly the state equation of the ideal gas for the first and second state: = RT, and = RT, that is: ( ) = R( T T) = RΔ T Inserting into the first law: f RΔ T = Q RΔ T B. Palásthy

11 f R R R Q = + Δ T = c + ΔT Introduce the heat at constant ressure as: f R c = +, this quantity is different for different gases. The heat: Q = c Δ T The sae way as before the olar secific heat is defined as: f C c = = + R. Note that in all cases C is greater than C, because at constant volue all heat absorbed is stored as internal energy, but at constant ressure the heat absorbed is stored as the change of the internal energy and soe additional work is done. R c = c +, or C = C + R The above stateent is called Robert-ayer equation. The ratio of the constant ressure secific heat to the constant volue secific heat is called adiabatic exonent: f + c C R γ =, or f + γ = = =. c f C R f.6. Isotheral rocess The teerature is ket constant: T = constant T = constant = RT = constant, =. ly the first law of therodynaics: Δ E = Q + W. In case of ideal gas the change of internal energy: B. Palásthy

12 f R Δ E = Δ T =, = Q + W W = Q W ' = Q The energy entering the syste as heat Q is equal to the work done by the syste W '. Deterine the work done by the environent, aly the definition of work done: W = d = d = ln we used that: =..6.4 diabatic rocess n adiabatic rocess is one in which there is no heat transfer either into or out of a syste; in other words: Q = Such a syste is called therally insulted. lying the first law: Δ E = Q + W Δ E = W, In eleentary for: de = δw Use the other for of the internal energy of an ideal gas: f E = The infinitesial change is: f f de = d + d, therefore: f f d + d = d, we have alied the eleentary work: δ W = d f f + d = d d f + d = f The adiabatic exonent is: C f + γ = =, C f d d = γ Taking the integral between two states: B. Palásthy

13 finally: d d = γ ln = γ ln, γ ln = ln γ =, =, γ = constant. γ γ This equation is called Poisson equation. Using the state equation as: = nrt, nrt = T nr γ = constant, therefore the Poisson equation between T and is: T γ = constant. B. Palásthy

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

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