77 SSM Figure 19-28 shows a hy-pothetical speed distribution forparticles of a certain gas: P(v) = Cv2for 0Vg. Find (a) an expression for C interms of vo, (b) the average speed ofthe particles, and (c) their rms speed.%3DVoSpeedFigure 19-28 Problem 77.(4)d

Given: The speed distribution is P(v) = Cv2. For 0 < v ≤vo. And P(v) = 0. For v > vo.

Answered: 77 SSM Figure 19-28 shows a hy-…

Physics for Scientists and Engineers: Foundations and Connections

1st Edition

ISBN:9781133939146

Author:Katz, Debora M.

Publisher:Katz, Debora M.

Chapter20: Kinetic Theory Of Gases

Section: Chapter Questions

Problem 44PQ

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The product of the pressure and volume of a sample of hydrogen gas at 0.00 is 80.0 J. (a) How many moles of hydrogen are present? (b) What is the average translational kinetic energy of the hydrogen molecules? (c) What is the value of the product of pressure and volume at 200 ?
Consider the Maxwell-Boltzmann distribution function plotted in Problem 28. For those parameters, determine the rms velocity and the most probable speed, as well as the values of f(v) for each of these values. Compare these values with the graph in Problem 28. 28. Plot the Maxwell-Boltzmann distribution function for a gas composed of nitrogen molecules (N2) at a temperature of 295 K. Identify the points on the curve that have a value of half the maximum value. Estimate these speeds, which represent the range of speeds most of the molecules are likely to have. The mass of a nitrogen molecule is 4.68 1026 kg. Equation 20.18 can be used to find the rms velocity given the temperature, Boltzmanns constant, and the mass of the atom or molecule. The mass of a nitrogen molecule is 4.68 1026 kg. vrms=3kBTm=3(1.381023J/K)4.681026kg=511m/s Using the results of Problem 28 and the rms velocity, we can calculate the value of f(v). f(vrms) = (3.11 108)(511)2 e(5.75106(511)2) = 0.00181 The most probable speed, for which this function has its maximum value, is given by Equation 20.20. vmp=2kBTm=2(1.381023J/K)(295K)4.681026kg=417m/s f(vmp) = (3.11108)(417)2 e(5.75106(417)2) = 0.00199 We plot these points on the speed distribution. The most probable speed is indeed at the peak of the distribution function. Since the function is not symmetric, the rms velocity is somewhat higher than the most probable speed. Figure P20.29ANS
Two monatomic ideal gases A and B are at the same temperature. If 1.0 g of gas A has the same internal energy as 0.10 g of gas B, what are (a) the ratio of the number of moles of each gas and (b) the ration of the atomic masses of the two gases?
An ideal gas expands quasi-statically and isothermally from a state with pressure p and volume V to a state with volume 4V. Show that the work done by the gas in the expansion is pV(ln 4).
Under what circumstances would you expect a gas to behave significantly differently than predicted by the ideal gas law?
Eight bumper cars, each with a mass of 322 kg. are running in a room 21.0 m long and 130 m wide. They have no driver, so they just bounce around on their own. The rms speed of the cars is 2.50 m/s. Repeating the arguments of Pressure, Temperature, and RMS Speed, find the average force per unit length (analogous to Pressure) that the cars exert the walls.
a) At what temperature do oxygen molecules have the same average speed as helium atoms ( M=4.00 g/mol) have at 300 K? b) What is answer to the same question about most probable speeds? c) What is the answer to the same question about rms speeds?
Average atomic and molecular speeds (vrms) are large, even at low temperatures. What is vrms for helium atoms at 5.00 K, just one degree above helium's liquefaction temperature?
From the MaxwellBoltzmann speed distribution, show that the most probable speed of a gas molecule is given by Equation 16.23. Note: The most probable speed corresponds to the point at which the slope of the speed distribution curve dNv/dv is zero.
A mole of gas has isobaric expansion coefficient dV/dT=R/p and isochoric pressure-temperature coefficient dp/dT=p/T . Find the equation of state of the gas.
A tank contains 111.0 g chlorine gas l2), which is at temperature 82.0 and absolute pressure 5.70105 Pa. The temperature of the air outside the tank is 20.0 . The molar mass of Cl2 is 70.9 g/mol. (a) What is the volume of the tank? (b) What is the internal energy of the gas? (c) What is the work done by the gas if the temperature and pressure inside the tank drop to 31.0 and 3.80105 Pa, respectively, due to a leak?
The pressure gauge on a cylinder of gas registers the gauge pressure, which is the difference between the interior pressure and the exterior pressure P0. Lets call the gauge pressure Pg. When the cylinder is full, the mass of the gas in it is mi at a gauge pressure of Pgi. Assuming the temperature of the cylinder remains constant, show that the mass of the gas remaining in the cylinder when the pressure reading is Pgf is given by mf=mi(Pgf+P0Pgi+P0)

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77 SSM Figure 19-28 shows a hy- pothetical speed distribution for particles of a certain gas: P(v) = Cv2 for 0 Vg. Find (a) an expression for C in terms of vo, (b) the average speed of the particles, and (c) their rms speed. %3D Vo Speed Figure 19-28 Problem 77. (4)d