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Pressure and Kinetic Energy,Assume a container is a cubeEdges are length _Look at the motion of the molecule in terms of its velocity componentsLook at its momentum and the average force,Pressure and Kinetic Energy, 2,Assume perfectly _ collisions with the walls of the containerThe relationship between the pressure and the molecular kinetic energy comes from momentum and Newtons Laws,Pressure and Kinetic Energy, 3,The relationship is This tells us that pressure is proportional to the number of molecules per unit volume (N/V) and to the average translational kinetic energy of the molecules,Pressure and Kinetic Energy, final,This equation also relates the macroscopic quantity of pressure with a microscopic quantity of the average value of the square of the molecular speedOne way to increase the pressure is to increase the number of _ per unit volumeThe pressure can also be increased by increasing the speed (kinetic energy) of the molecules,Molecular Interpretation of Temperature,We can take the pressure as it relates to the kinetic energy and compare it to the pressure from the equation of state for an ideal gasTherefore, the temperature is a direct measure of the average molecular kinetic energy,Molecular Interpretation of Temperature, cont,Simplifying the equation relating temperature and kinetic energy givesThis can be applied to each _, with similar expressions for vy and vz,A Microscopic Description of Temperature, final,Each translational degree of freedom contributes an _ amount to the energy of the gasIn general, a degree of freedom refers to an independent means by which a molecule can possess energyA generalization of this result is called the theorem of equipartition of energy,Theorem of Equipartition of Energy,Each degree of freedom contributes _kBT to the energy of a system, where possible degrees of freedom in addition to those associated with translation arise from rotation and vibration of molecules,Total Kinetic Energy,The total kinetic energy is just N times the kinetic energy of each moleculeIf we have a gas with only translational energy, this is the internal energy of the gasThis tells us that the internal energy of an ideal gas depends only on the temperature,Root Mean Square Speed,The root mean square (rms) speed is the square root of the average of the squares of the speedsSquare, average, take the square rootSolving for vrms we findM is the molar mass and M = mNA,Some Example vrms Values,At a given temperature, _ molecules move faster, on the average, than heavier molecules,Equipartition of Energy,With complex molecules, other contributions to internal energy must be taken into accountOne possible energy is the _ motion of the center of mass,Equipartition of Energy, 2,_ motion about the various axes also contributesWe can neglect the rotation around the y axis since it is negligible compared to the x and z axes,Equipartition of Energy, 3,The molecule can also _ There is kinetic energy and potential energy associated with the vibrations,Equipartition of Energy, 4,The translational motion adds _ degrees of freedomThe rotational motion adds two degrees of freedomThe vibrational motion adds two more degrees of freedomTherefore, Eint = 7/2 nRT and CV = 7/2 RThis is inconsistent with experimental results,Agreement with Experiment,Molar specific heat is a function of _ At low temperatures, a diatomic gas acts like a monatomic gasCV = 3/2 R,Agreement with Experiment, cont,At about room temperature, the value increases to CV = 5/2 RThis is consistent with adding rotational energy but not vibrational energyAt high temperatures, the value increases to CV = 7/2 RThis includes _ energy as well as rotational and translational,Complex Molecules,For molecules with more than two atoms, the vibrations are more complexThe number of degrees of freedom is largerThe more degrees of freedom available to a molecule, the more “ways” there are to store energyThis results in a higher molar specific _,Quantization of Energy,To explain the results of the various molar specific heats, we must use some quantum mechanicsClassical mechanics is not sufficientIn quantum mechanics, the energy is proportional to the frequency of the wave representing the frequencyThe _ of atoms and molecules are quantized,Quantization of Energy, 2,This energy level diagram shows the rotational and vibrational states of a diatomic moleculeThe lowest allowed state is the _ state,Quantization of Energy, 3,The vibrational states are separated by larger energy gaps than are rotational statesAt low temperatures, the energy gained during collisions is generally not enough to raise it to the first excited state of either _ or vibration,Quantization of Energy, 4,Even though rotation and vibration are classically allowed, they do not occurAs the temperature increases, the energy of the molecules _ In some collisions, the molecules have enough energy to excite to the first excited stateAs the temperature continues to increase, more molecules are in excited states,Quantization of Energy, final,At about room temperature, _ energy is contributing fullyAt about 1000 K, vibrational energy levels are reachedAt about 10 000 K, vibration is contributing fully to the internal energy,Boltzmann Distribution Law,The motion of molecules is extremely chaoticAny individual molecule is colliding with others at an enormous rateTypically at a rate of a billion times per secondWe add the number density nV (E )This is called a distribution functionIt is defined so that nV (E ) dE is the number of molecules per unit volume with energy between E and E + _,Number Density and Boltzmann Distribution Law,From statistical mechanics, the number density is nV (E ) = noe E /_TThis equation is known as the Boltzmann distribution lawIt states that the probability of finding the molecule in a particular energy state varies exponentially as the energy divided by kBT,Distribution of Molecular Speeds,The observed _ distribution of gas molecules in thermal equilibrium is shown at rightNV is called the Maxwell-Boltzmann speed distribution function,Distribution Function,The fundamental expression that describes the distribution of speeds in N gas molecules is m is the mass of a gas molecule, _ is Boltzmanns constant and T is the absolute temperature,Most Probable Speed,The average speed is somewhat _ than the rms speedThe most probable speed, vmp is the speed at which the distribution curve reaches a peak,Speed Distribution,The peak shifts to the _ as T increasesThis shows that the aver

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