Example: tourism industry

Temperature ~ Average KE of each particle Gas Particles ...

Temperature ~ Average KE of each particle Particles have different speeds Gas Particles are in constant RANDOM motion Average KE of each particle is: 3/2 kT Pressure is due to momentum transfer Speed Distribution at CONSTANT Temperature is given by the Maxwell Boltzmann Speed Distribution Pressure and Kinetic Energy Assume a container is a cube with edges d. Look at the motion of the molecule in terms of its velocity components and momentum and the Average force Pressure is proportional to the number of molecules per unit volume (N/V) and to the Average translational kinetic energy of the molecules. This equation also relates the macroscopic quantity of pressure with a microscopic quantity of the Average value of the square of the molecular speed One way to increase the pressure is to increase the number of molecules per unit volume The pressure can also be increased by increasing the speed (kinetic energy) of the molecules ___22132oNPmvV Molecular Interpretation of Temperature We can take the pressure as

Temperature ~ Average KE of each particle • Particles have different speeds • Gas Particles are in constant RANDOM motion • Average KE of each particle is: 3/2 kT • Pressure is due to momentum transfer Speed ‘Distribution’ at CONSTANT Temperature is given by the

Tags:

  Temperatures, Particles, Average, Each, Average ke of each particle

Information

Domain:

Source:

Link to this page:

Please notify us if you found a problem with this document:

Other abuse

Transcription of Temperature ~ Average KE of each particle Gas Particles ...

1 Temperature ~ Average KE of each particle Particles have different speeds Gas Particles are in constant RANDOM motion Average KE of each particle is: 3/2 kT Pressure is due to momentum transfer Speed Distribution at CONSTANT Temperature is given by the Maxwell Boltzmann Speed Distribution Pressure and Kinetic Energy Assume a container is a cube with edges d. Look at the motion of the molecule in terms of its velocity components and momentum and the Average force Pressure is proportional to the number of molecules per unit volume (N/V) and to the Average translational kinetic energy of the molecules. This equation also relates the macroscopic quantity of pressure with a microscopic quantity of the Average value of the square of the molecular speed One way to increase the pressure is to increase the number of molecules per unit volume The pressure can also be increased by increasing the speed (kinetic energy)

2 Of the molecules ___22132oNPmvV 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 gas Temperature is a direct measure of the Average molecular kinetic energy ___2B2132 NPmvnRTNkTVMolecular Interpretation of Temperature Simplifying the equation relating Temperature and kinetic energy gives This can be applied to each direction, with similar expressions for vy and vz ___2B1322omvkT ___2B1122xmvkT Total Kinetic Energy The total kinetic energy is just N times the kinetic energy of each molecule If we have a gas with only translational energy, this is the internal energy of the gas This tells us that the internal energy of an ideal gas depends only on the Temperature ___2tot transB133222 KNmvNkTnRT Kinetic Theory Problem A vessel contains nitrogen gas at C and atm.

3 Find (a) the total translational kinetic energy of the gas molecules and (b) the Average kinetic energy per molecule. Hot Question Suppose you apply a flame to 1 liter of water for a certain time and its Temperature rises by 10 degrees C. If you apply the same flame for the same time to 2 liters of water, by how much will its Temperature rise? a) 1 degree b) 5 degrees c) 10 degrees d) zero degrees Ludwig Boltzmann or Dean Gooch? 1844 1906 Austrian physicist Contributed to Kinetic Theory of Gases Electromagnetism Thermodynamics Pioneer in statistical mechanics Distribution of Molecular Speeds The observed speed distribution of gas molecules in thermal equilibrium is shown at right NV is called the Maxwell-Boltzmann speed distribution function mo is the mass of a gas molecule, kB is Boltzmann s constant and T is the absolute Temperature 23/ 2/22B42 BmvkToVmNNvekT Molecular Speeds and Collisions Speed Summary Root mean square speed The Average speed is somewhat lower than the rms speed The most probable speed.

4 Vmp is the speed at which the distribution curve reaches a peak vrms > vavg > vmp Example vrms Values At a given Temperature , lighter molecules move faster, on the Average , than heavier molecules Speed Distribution The peak shifts to the right as T increases This shows that the Average speed increases with increasing Temperature The asymmetric shape occurs because the lowest possible speed is 0 and the highest is infinity 23 / 21/ 2rmskTKEmv 23rmskTvvm Root-mean-square speed: The Kelvin Temperature of an ideal gas is a measure of the Average translational kinetic energy per particle : k = x 10-23 J/K Boltzmann s Constant Kinetic Theory Problem Calculate the RMS speed of an oxygen molecule in the air if the Temperature is C.

5 The mass of an oxygen molecule is u (k = 8x 10 -23 J/K, u = x 10 -27 kg) 3rmskTvm What is m? m is the mass of one oxygen molecule in kg. What is u? How do we get the mass in kg? Kinetic Theory Problem Calculate the RMS speed of an oxygen molecule in the air if the Temperature is C. The mass of an oxygen molecule is u (k = 8x 10 -23 J/K, u = x 10 -27 kg) 3rmskTvm 23273( 10/ )278(32 )( 10/ )xJ KKuxkg u 466 /ms What is m? m is the mass of one oxygen molecule. Is this fast? YES! Speed of sound: 343m/s! A cylinder contains a mixture of helium and argon gas in equilibrium at 150 C. (a) What is the Average kinetic energy for each type of gas molecule? (b) What is the root-mean-square speed of each type of molecule?

6 More Kinetic Theory Problems A gas molecule with a molecular mass of u has a speed of 325 m/s. What is the Temperature of the gas molecule? A) K B) 136 K C) 305 K D) 459 K E) A Temperature cannot be assigned to a single molecule. Temperature ~ Average KE of all Particles Equipartition of Energy each translational degree of freedom contributes an equal amount to the energy of the gas In general, a degree of freedom refers to an independent means by which a molecule can possess energy each degree of freedom contributes kBT to the energy of a system, where possible degrees of freedom are those associated with translation, rotation and vibration of molecules With complex molecules.

7 Other contributions to internal energy must be taken into account One possible energy is the translational motion of the center of mass Rotational motion about the various axes also contributes There is kinetic energy and potential energy associated with the vibrations Monatomic and Diatomic Gases The thermal energy of a monatomic gas of N atoms is A diatomic gas has more thermal energy than a monatomic gas at the same Temperature because the molecules have rotational as well as translational kinetic energy. Molar Specific Heat We define specific heats for two processes that frequently occur: Changes with constant pressure Changes with constant volume Using the number of moles, n, we can define molar specific heats for these processes Molar specific heats.

8 Q = nCV DT for constant-volume processes Q = nCP DT for constant-pressure processes Ideal Monatomic Gas Therefore, DEint = 3/2 nRT DE is a function of T only In general, the internal energy of an ideal gas is a function of T only The exact relationship depends on the type of gas At constant volume, Q = DEint = nCV DT This applies to all ideal gases, not just monatomic ones Ratio of Molar Specific Heats We can also define the ratio of molar specific heats Theoretical values of CV , CP , and g are in excellent agreement for monatomic gases But they are in serious disagreement with the values for more complex molecules Not surprising since the analysis was for monatomic gases 5 /2 PVCRCRg Agreement with Experiment Molar specific heat is a function of Temperature At low temperatures , a diatomic gas acts like a monatomic gas CV = 3/2 R At about room Temperature , the value increases to CV = 5/2 R This is consistent with adding rotational energy but not vibrational energy At high temperatures , the value increases to CV = 7/2 R This includes vibrational energy as well as rotational and translational Sample Values of Molar Specific Heats In a constant-volume process.

9 209 J of energy is transferred by heat to mol of an ideal monatomic gas initially at 300 K. Find (a) the increase in internal energy of the gas, (b) the work done on it, and (c) its final Temperature Molar Specific Heats of Other Materials The internal energy of more complex gases must include contributions from the rotational and vibrational motions of the molecules In the cases of solids and liquids heated at constant pressure, very little work is done, since the thermal expansion is small, and CP and CV are approximately equal Adiabatic Processes for an Ideal Gas An adiabatic process is one in which no energy is transferred by heat between a system and its surroundings (think styrofoam cup)

10 Assume an ideal gas is in an equilibrium state and so PV = nRT is valid The pressure and volume of an ideal gas at any time during an adiabatic process are related by PV g = constant g = CP / CV is assumed to be constant All three variables in the ideal gas law (P, V, T ) can change during an adiabatic process Adiabatic Process The PV diagram shows an adiabatic expansion of an ideal gas The Temperature of the gas decreases Tf < Ti in this process For this process Pi Vig = Pf Vfg and Ti Vig-1 = Tf Vfg-1 A sample of a diatomic ideal gas expands slowly and adiabatically from a pressure of atm and a volume of L to a final volume of L. (a) What is the final pressure of the gas?


Related search queries