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1、Chapter 7 The Kinetic Theory of Gases 7-1 Temperature and Thermal Equilibrium 7-2 Thermodynamic Variables and the Equation of State 7-3 Pressure and Molecular Motion 7-4 The Microscopic Interpretation of Temperature 7-5 The Equipartition of Energy 7-6 Mean Free Path 7-7 The Distribution of Molecular
2、 Speeds 7-8 The Boltzmann Distribution 7-9 Van der Waals Equation of State,Thermal physics is the study of properties and motions laws of matters that are temperature dependent.Thermodynamics studies thermal phenomena from macroscopic point of view,such as seeks to establish quantitative relationshi
3、ps among macroscopic variables(p,V,T,)of thermal systems,or investigates thermal(or internal)energy transfers involving temperatures between two macroscopic bodies.Statistical physics studies thermal phenomena from macroscopic point of view based on microscopic structure of matter and the mechanical
4、 laws abided by each microscopic particle.,7-1 Temperature and Thermal Equilibrium,Thermal equilibrium:,If two objects at different temperatures are placed in thermal contact(meaning thermal energy can pass from one to the other),the two objects will eventually reach the same temperature.They will t
5、hen be said to be in thermal equilibrium.,The zeroth law of thermodynamics.If bodies A and B,are each in thermal equilibrium with a third body C,then they are in thermal equilibrium with each other.,Celsius scale:Celsius temperature TC=T273.15,Kelvin scale:Kelvin temperature T=TC+273.15K,Temperature
6、 is one of the seven SI base quantities:,Fahrenheit scale,7-2 Thermodynamic Variables and the Equation of State,We call a physical system a thermodynamic system(or system)whenever we are interested in its temperature-dependent properties.Everything else in the universe we will refer to as the enviro
7、nment or surroundings of the system.,The variables that describe a thermodynamic system,such as a bottle of gas with a piston,are the pressure p,temperature T,volume V,and the number of moles.These variables are called thermodynamic(or state)variables.Equilibrium state is a state of a system in whic
8、h all the thermodynamic variables have definite values that remain constant so long as the external conditions are unchanged.,Ideal Gas Law,Here,p,V,and T stand for pressure,volume,and temperature,respectively;m is the mass of gas present;M is the molar mass(the mass of 1 mole);m/M is the number of
9、moles of gas present.,R=8.31 J/molK.,(the universal gas constant),An alternative form of ideal gas law:,The Boltzmanns constant,Let n=N/V be the number density of the gas molecules,we get,or,Checkpoint 1,7-3 Pressure and Molecular Motion,The principal assumptions of the kinetic theory of gases,A sam
10、ple of gas consists of many identical molecules.The molecules are very far apart in comparison to their size;The direction of motion of any molecule is random;The molecules are treated as if they were hard spheres.The molecules obey Newtons laws of motion.,These assumptions constitute the definition
11、 of an ideal gas.Any gas that obeys the relations derived from these assumptions at all temperatures and pressures is called an ideal gas.,Pressure,Consider an ideal gas confined to a cubical box with sides of length L.The molecules of gas are moving in all directions and with various speeds.,A typi
12、cal gas molecule:m,Assume that any collision of a molecule with a wall is elastic,when this molecule collides with the shaded wall,the change of the molecules momentum is,collides with the right wall.,The impulse delivered to the wall by the moleculeduring the collision is,I=px=+2mvx,The average rat
13、e at which impulse is delivered to the shaded wall by this single molecule is,From Newtons second law(Fx=dpx/dt),the force acting on that wall due to this single molecule Fix=mvix2/L,Total force is the sum of all the molecules contribution:,Pressure p on that wall,For any molecule,v2=vx2+vy2+vz2,and
14、 molecules are all moving in random directions,therefor,It tells us how the pressure of the gas(apurely macroscopic quantity)depends onthe speed of the molecules(a purely microscopic quantity),or,7-4 The Microscopic Interpretation of Temperature,We can rewrite,as,here n=N/V.,per molecule associated
15、with random molecular motion.,the average translational kinetic energy,The average translational kinetic energy of molecules in an ideal gas directly depends only on the temperature,not on the pressure or type of gas.,We get,provides a new definition of temperature in terms of the microscopic proper
16、ties of a gas.Specifically,temperature is a measure of the average random translational kinetic energy of the molecules of a gas,We can calculate how fast molecules are moving on average:,Root-mean-square speed,Checkpoint 2,Example 7-2 What are the average translational kinetic energy and the rms sp
17、eed of a nitrogen molecule at a temperature of 300 K?Assume that the nitrogen behaves as an ideal gas.The mass of a nitrogen molecule is m=4.651026 kg.,Solution:,At room temperature,the rms speed of nitrogen molecules is 517 m/s.The speed of sound in this gas at this temperature is 350 m/s.Why?,7-5
18、The Equipartition of Energy,A classical molecule in thermal equilibrium at temperature T has an average energy of kT/2 for each independent mode of motion or so-called degree of freedom.,Consider translational motion of a molecule:,3 degrees of freedom corresponding to translational motion of the mo
19、lecule along the x,y,and z directions,Equipartition theorem,The total average transformational kinetic energy,This is agreement with,Degrees of Freedom,Monatomic molecule f=3(translational),Diatomic molecule f=5(plus two rotational),Polyatomic molecule f=6(plus three rotational),The number of indepe
20、ndent modes of motion:,A degree of freedom is also associated with rotational velocity as well:,Suppose that the number of degrees of freedom of a molecule is f,from the equipartition of energy,the average kinetic energy for each molecule is given by,Average Kinetic Energy,For a monatomic molecule,f
21、=3,For a diatomic molecule,f=5,For a polyatomic molecule,f=6,Internal Energy of Ideal Gases,The internal energy Eint of an ideal gas is simply the sum of the kinetic energies of its molecules(or atoms for a monatomic ideal gas)because for an ideal gas no forces act between molecules,which means the
22、potential energy between molecules is zero.,Using k=R/NA and the number of moles=m/M,we can rewrite this as,Avogadros number NA(=6.021023),7-6 Mean Free Path,Molecules,even in a dilute ideal gas,undergo a vast number of collisions with each other about 5109/s in the air around us,which is about 105
23、collisions for each centimeter of path traveled.,The figure shows the path of a typical molecule as it moves through an ideal gas,changing both speed and direction abruptly as it collides elastically with other molecules.,distance traveled by a molecule between collisions,Mean free path:the average,
24、n is number density of the molecules.d is the molecular diameter.,Assume that a typical molecule“A”is traveling with aconstant speed v and that all the other molecules are at rest.We assume further that the molecules are spheres,of diameter d.A collision will then take place if the centers of molecu
25、les come within a distance d of each other.Namely,centers of molecules located within the zigzag cylinder will collide with molecule A.,In fact,all molecules are moving;when this is taken properly into account,Checkpoint 3,Example 7-3(a)What is the mean free path for oxygen molecules at temperature
26、T=300 K and pressure p=1.01105 Pa?Assume that the molecular diameter is d=290 pm and the gas is ideal.,Solution:,From,We get,and,=1.1107 m,This is about 380 molecular diameters.,(a),(b)Assume the average speed of the oxygen molecules is v=500 m/s.What is the average time t between successive collisi
27、ons for any given molecule?At what rate does the molecule collide;that is,what is the frequency f of its collisions?,Solution:,The average time between collisions is,The average rate or frequency f at which the collisions occur is the inverse of the time t between collisions:,This tells us that,on a
28、verage,any given oxygen molecule makes about 5 billion collisions per second.,7-7 The Distribution of Molecular Speeds,The root-mean-square speed vrms gives us a general idea of molecular speeds in a gas at a given temperature.We often want to know more.For example,what fraction of the molecules hav
29、e speeds greater than the rms value?Greater than twice the rms value?,According to the classical mechanics theory,the molecular speeds in a gas can have continuous values from zero to infinity.Therefore we need to group molecules according to their speed ranges,say,with 10 m/s speed interval to divi
30、de all possible speeds into 010,1020,2030 m/s,of speed ranges,and then describe how many molecules lie in each individual speed ranges.,The Speed Distribution Function,Consider a gas with total number of N molecules.Suppose the number of molecules with speeds betweenv and v dv is dN,then the speed d
31、istribution function is defined as,The physical meaning of the speed distributionfunction is the fraction of number of molecules per unit speed interval whose speeds centered on speed v.,The quantity f(v)is a probability distribution function.,is the fraction of molecules whose speeds lie in the int
32、erval of width dv centered on speed v.In other words,f(v)dv is the probability that a gas molecules speed is between v and v dv.,Maxwells Speed Distribution Law,In 1859,Scottish physicist James Clerk Maxwell worked out the speed distribution function of ideal gas molecules:,f(v)dv=the fraction of mo
33、lecules whose speeds lie in the interval of width dv centered,on speed v,or the probability that a gas molecules speed is between v and vdv.,the fraction of molecules whose speeds lie in v1 to v2:,The total area under the curve,Average,RMS,and Most Probable Speeds,Average speed,Substituting for f(v)
34、and taking the definite integral,Root-mean-square speed,The average of the square of the speeds,Substituting for f(v),we have,The most probable speed vp,is the speed at which f(v),is maximum,Set df(v)/dv=0 and then solve for v:,The curve changes with the temperatures or molar masses of gases.,If T2
35、T1 or M2M1.to keep the area is unity under the curve.,Checkpoint 4,Summary,Example 7-4 Suppose a sample of helium gas at temperature T=300 K contains N=106 atoms,each of mass m=6.651027 kg.(a)What is the most probable speed vp of a helium atom in the sample?,Solution:,(b)Estimate how many atoms in t
36、he sample have speeds in the range between vp and vp 40 m/s.,Solution:From the definition of the speed distribution,function,the number of molecules having a speed between v and v v is,3104,About 3%of the helium molecules will have speeds that lie in the range between 1120 m/s and 1160 m/s.,(c)Now l
37、et v=10 vp,and calculate the number of molecules between v and v 40 m/s.,Solution:,Set v=10vp=1.12104 m/s,we obtain,N=41039.,This means essentially that no molecules have speeds this high.,7-8 The Boltzmann Distribution,The velocity distribution function,Suppose the number of molecules with the velo
38、city components between vx and vx dvx,vy and vy dvy,vz and vz dvz,is dN,is the fraction of molecules whose velocities lie in the,interval of width dvxdvydvz centered on velocity,In other words,F(v)dvxdvydvz is the probability that,a gas molecules velocity is between and,Get the velocity distribution
39、 function from the Maxwells speed distribution law,4v2dv.,F(v)4v2dv.,dvxdvydvz represents a tiny box at the tip of a vector,whose length is the speed v.We can sum(integrate)over all possible directions,keeping the speed fixed.,Because F(v)4v2dv is got by Summing(integrating)over all possible directi
40、ons,keeping the speed fixed,it represents a speed distribution,F(v)4v2dv=f(v)dv,Substituting for f(v)from Maxwells speed distribution law,we have,Boltzmann generalized the velocity distribution to a distribution that describes the probability that any one molecule has a given energy E.He proposed th
41、at in,the original velocity distribution,the total energy of the molecule,E,should replace the kinetic energy mv2/2,Boltzmann Distribution,Let us apply the Boltzmann distribution to find the number of molecules per unit volume as a function of altitude in the gravitational filed,The total energy of
42、a molecule,in the state range of dvxdvydvzdxdydz,is given by,From the Boltzmann distribution,the number of molecules lies in that state range is,By integrating over all possible velocities,we can get the number of molecules which lies in the differential volume element dxdydz,The number density of m
43、olecules,If we substitute n0(the number of density at z=0)for C,h(the altitude of the molecular position above the Earth surface)for z,This is known as the altitude distribution law of molecules or particles in the gravitational field.It gives the number density of particles as a function of altitud
44、e h.,Pressure as a function of altitude,Multiplied by kT for both sides,p0=n0kT is the pressure at h=0.,which means that the atmospheric pressure will be decreased by 133 Pa for every 10 m altitude increase.This is the principle for a kind of altimeter.,7-9 Van der Waals Equation of State,The ideal
45、gas law is an accurate description of the behavior of a real gas as long as the pressure is not too high and the temperature is far from the liquefaction point.But what happens when these two criteria are not satisfied?In other words,we need to find an equation of state to describe the behaviors of
46、real gases.,In 1873 Johannes D.van der Waals proposed an equation of state that works well for many real gases.His formula,known as the van der Waals equation,The constants a and b are both positive.,Interpretation of the van der Waals Gas,If we solve for p,Explain:,them over distances larger than t
47、heir radii.These forces are repulsive at short range,so that the molecules bounce off one another,but are slightly attractive at long distances.The attractive component of intermolecular force acts to make a gas more compact.This translates into a reduced pressure.The second term represents the effe
48、ct of the long-range attraction between molecules,Real molecules have forces that act between,As for the term proportional to b,molecules,like hard spheres,take up some space.This term is present because of the strong repulsion between molecules at a characteristic radius.By appearing as a term subtracted from V,the constant b measures the volume unavailable for the motion of molecules because the volume is already occupied by other molecules.,
