Sox Red - Yankees. Baseball can not get more exciting than these games.
Frank L. H. Wolfs
Department of Physics and Astronomy, University of Rochester
Physics 121, April 17, 2008. Kinetic theory of gases.
http://eml.ou.edu/Physics/module/thermal/ketcher/Idg4.avi
Frank L. H. Wolfs Department of Physics and Astronomy, University of Rochester
Physics 121. April 17, 2008.
Course Information Topics to be discussed today:
The ideal gas law (review). Molecular interpretation of temperature. The real equation of state.
Quiz
Frank L. H. Wolfs
Department of Physics and Astronomy, University of Rochester
Physics 121. April 17, 2008.
Homework set # 9 is due on Saturday morning, April 19, at 8.30 am. Midterm Exam # 3 will take place on Tuesday April 22 between 8.00 am and 9.30 am. The material to be covered is the material contained in Chapters 10, 11, 12, and 14. I will distribute the formula sheet that will be attached to the exam later this week via email. I will review the material covered on the exam on Sunday evening. Further details will be announced via email. Extra ofce hours will be offered by the TAs on Sunday and Monday. The schedule will be announced via email. There will be no class after the exam on Tuesday.
Frank L. H. Wolfs Department of Physics and Astronomy, University of Rochester
Physics 121. April 17, 2008.
Workshops schedule next week:
There will be workshops on Monday to answer questions related to Exam # 3. There will be workshops on Thursday and Friday to discuss the solutions of Exam # 3 and return your blue booklets. During the week of April 28 there will only be workshops on Monday, Tuesday, and Wednesday.
To increase the efciency with which we can return the blue booklets of Exam # 3, you will need to write your workshop day/time on the front of each booklet. The results of Exam # 3 will be emailed to you on Monday April 28
Frank L. H. Wolfs Department of Physics and Astronomy, University of Rochester
Physics 121. April 17, 2008.
Grades: Worst case scenario:
Lets assume you got 0% on exam 1 and 0% on exams 2. Clearly you would benet if we drop one of these grades! If we drop one of these grades, than the one we count will count for 20% of your nal grade. If you get 100% on exam # 3 and 100% on the nal (and assuming you have 100% on the homework sets, quizzes, and labs) you end up with a nal grade for 80% or an A!!!!! No one matches this worst case scenario, but realize that even if you did poorly on the rst two exams, you can still get an A in this course.
Frank L. H. Wolfs
Department of Physics and Astronomy, University of Rochester
The equation of state of a gas. A quick review.
The equation of state of a gas was initially obtained on the basis of observations.
Boyles Law (1627 - 1691): pV = constant for gases maintained at constant temperature. Charles Law (1746 - 1823): V/T = constant for gases maintained at constant pressure. Gay-Lussacs Law (1778 - 1850): p/T = constant for gases maintained at constant volume
Frank L. H. Wolfs Department of Physics and Astronomy, University of Rochester
The equation of state of a gas. A quick review.
The equation of state of a gas can be written as pV = nRT where
p = pressure (in Pa). V = volume (in m3). n = number of moles of gas (1 mole = 6.02 x 1023 molecules or atoms). Note the number of molecules in a mole is also known as Avogadros number NA. R = the universal gas constant (R = 8.315 J/(mol K). T = temperature (in K).
Note: the equation of state is the equation of state of an ideal gas. Gases at very high pressure and/or close to the freezing point show deviations from the ideal gas law.
Frank L. H. Wolfs Department of Physics and Astronomy, University of Rochester
The equation of state of a gas. A quick review.
The equation of state of a gas can also be written as pV = nRT = (N/NA)RT = NkT where
N = total number of molecules NA = Avogadros number k = R/ NA is the Boltzmann constant = 1.38 x 10-23 J/K
This is the most frequently used form of the equation of state of the gas.
Frank L. H. Wolfs
Department of Physics and Astronomy, University of Rochester
The molecular point of view of a gas.
Consider a gas contained in a container. The molecules in the gas will continuously collide with the walls of the vessel. Each time a molecule collides with the wall, it will carry out an elastic collision. Since the linear momentum of the molecule is changed, the linear momentum of the wall will change too. Since force is equal to the change in linear momentum per unit time, the gas will exert a force on the walls.
Frank L. H. Wolfs Department of Physics and Astronomy, University of Rochester
The molecular point of view of a gas.
Consider the collision of a single molecule with the left wall. In this collision, the linear momentum of the molecule changes by mvx - (-mvx) = 2mvx The same molecule will collide with this wall again after a time t = 2l/vx The force that this single molecule exerts on the left wall is thus equal to p/t = (2mvx)/(2l/vx) = mvx2/l
Frank L. H. Wolfs Department of Physics and Astronomy, University of Rochester
The molecular point of view of a gas.
The force that this single molecule exerts on the left wall is thus equal to Fleft = mvx2/l If the pressure exerted on the left wall this by molecule is equal to pleft = Fleft/A = mvx2/(lA) where A is the area of the left wall. The volume of the gas is equal to lA and we can thus rewrite the pressure on the left wall: pleft = mvx2/V
Frank L. H. Wolfs Department of Physics and Astronomy, University of Rochester
The molecular point of view of a gas.
The pressure that many molecules exerts on the left wall is equal to pleft = m(v1x2+v2x2+v3x2+... )/V This equation can be rewritten in terms of the average of the square of the x component of the molecular velocity and the number of molecules (N): pleft = mN(vx2)average/V Assuming that there is no preferential direction, the average square of the x, y, and z components of the molecular velocity will be the same: (vx2)average= (vy2)average= (vz2)average
Frank L. H. Wolfs Department of Physics and Astronomy, University of Rochester
The molecular point of view of a gas.
The force on the left wall can be rewritten in terms of the average squared velocity pleft = mN(v2)average/3V Assuming there is no preferential direction of motion of the molecules, the pressure on all walls will be the same and we thus conclude: pV = mN(v2)average/3 Compare this to the ideal gas law: pV = NkT
Frank L. H. Wolfs Department of Physics and Astronomy, University of Rochester
The molecular point of view of a gas.
Comparing pV = mN(v2)average/3 and the ideal gas law pV = NkT we nd that kT = m(v2)average/3 = (2/3)Kaverage We thus conclude that:
The average kinetic energy of the gas molecules is proportional to the temperature of the gas. When the temperature approaches zero, the average kinetic energy approaches zero.
Frank L. H. Wolfs Department of Physics and Astronomy, University of Rochester
Distribution of molecular speeds. The Maxwell distribution.
Fixed temperature
Integral = N
Most probable v
Average of v2 Average v (50% below, 50% above)
Frank L. H. Wolfs
Department of Physics and Astronomy, University of Rochester
Distribution of molecular speeds. The Maxwell distribution.
Note: the integral does not change when T changes!
Note: all velocities scale with T
Increasing T increases the fraction of molecules above a certain velocity.
Frank L. H. Wolfs Department of Physics and Astronomy, University of Rochester
Molecular speed and the mean-free path.
The RMS velocities of individual gas molecules are large. For example for hydrogen at room temperature, the RMS velocity is 1920 m/s. Despite the large RMS velocity, the average diffusion velocity is much smaller and is largely determined by the mean-free path of the molecules. We expect that the mean-free path is inversely proportional to the crosssectional are of the molecules and inversely proportional to the density.
Frank L. H. Wolfs
Typical values of the mean-free path are between 10-8 and 10-7 m
Department of Physics and Astronomy, University of Rochester
Molecular speed and the propagation of sound.
Typical values for vrms:
For H at 300 K, vrms = 1920 m/s For N at 300 K, vrms = 517 m/s
The speed of sound in these two gases is 1350 m/s for H and 350 m/s for N. Note: The speed of sound in a gas will always be less than vrms since the sound propagates thro...