How well does the ideal gas law describe the

(I) (a) What is the average translational kinetic energy of an oxygen molecule at STP? (b) What is the total translational kinetic energy of 1.0 mol of 0 2 molecules at 25C?

(I) Calculate the rms speed of helium atoms near the surface of the Sun at a temperature of about 6000 K.

(I) By what factor will the rms speed of gas molecules increase if the temperature is increased from 0C to 180C?

(I) A gas is at 20 C. To what temperature must it be raised to triple the rms speed of its molecules?

(I) What speed would a 1.0-g paper clip have if it had the same kinetic energy as a molecule at 15C?

(I) A 1.0-mol sample of hydrogen gas has a temperature of 27C. (a) What is the total kinetic energy of all the gas molecules in the sample? (b) How fast would a 65-kg person have to run to have the same kinetic energy?

(I) Twelve molecules have the following speeds, given in arbitrary units: 6.0, 2.0, 4.0, 6.0, 0.0, 4.0,1.0, 8.0, 5.0, 3.0, 7.0, and 8.0. Calculate (a) the mean speed, and (b) the rms speed.

(II) The rms speed of molecules in a gas at 20.0 C is to be increased by 2.0%. To what temperature must it be raised?

(II) If the pressure in a gas is tripled while its volume is held constant, by what factor does i;rms change?

(II) Show that the rms speed of molecules in a gas is given by ^rms = V3P /p , where P is the pressure in the gas, and p is the gas density.

(II) Show that for a mixture of two gases at the same temperature, the ratio of their rms speeds is equal to the inverse ratio of the square roots of their molecular masses.

(II) What is the rms speed of nitrogen molecules contained in an 8.5-m3 volume at 3.1 atm if the total amount of nitrogen is 1800 mol?

(II) (a) For an ideal gas at temperature T show that ^rms _ 1 ^rms dT 2 T and using the approximation Avrms ~ ^rms AT, show that A^rms _ 1 A T ^rms 2 T (b) If the average air temperature changes from -5C in winter to 25 C in summer, estimate the percent change in the rms speed of air molecules between these seasons.

(II) What is the average distance between oxygen molecules at STP?

(II) Two isotopes of uranium, 235U and 238U (the superscripts refer to their atomic masses), can be separated by a gas diffusion process by combining them with fluorine to make the gaseous compound UF6. Calculate the ratio of the rms speeds of these molecules for the two isotopes, at constant T. Use Appendix F for masses.

Can pockets of vacuum persist in an ideal gas? Assume that a room is filled with air at 20 C and that somehow a small spherical region of radius 1 cm within the room becomes devoid of air molecules. Estimate how long it will take for air to refill this reaion o f var.iinm. Assume the

"(II) Calculate (a) the rms speed of a nitrogen molecule at 0C and (b) determine how many times per second it would move back and forth across a 5.0-m-long room on the average, assuming it made very few collisions with other molecules."

(Ill) Estimate how many air molecules rebound from a wall in a typical room per second, assuming an ideal gas of N molecules contained in a cubic room with sides of length I at temperature T and pressure P. (a) Show that the frequency / with which gas molecules strike a wall is where vx is the average x component of the molecules velocity. (b) Show that the equation can then be written as p f V 4mAT where m is the mass of a gas molecule, (c) Assume a cubic air-filled room is at sea level, has a temperature 20 C, and has sides of length = 3 m. Determine / .

(I) If you double the mass of the molecules in a gas, is it possible to change the temperature to keep the velocity distribution from changing? If so, what do you need to do to the temperature?

(I) A group of 25 particles have the following speeds: two have speed 10 m/s, seven have 15 m/s, four have 20 m/s, three have 25 m/s, six have 30 m/s, one has 35 m/s, and two have 40 m/s. Determine (a) the average speed, (b) the rms speed, and (c) the most probable speed.

(II) A gas consisting of 15,200 molecules, each of mass 2.00 X 10 26 kg, has the following distribution of speeds, which crudely mimics the Maxwell distribution: Number of Molecules Speed (m/s) 1600 220 4100 440 4700 660 3100 880 1300 1100 400 1320 (a) Determine vrms for this distribution of speeds. (b) Given your value for vrms, what (effective) temperature would you assign to this gas? (c) Determine the mean speed v of this distribution and use this value to assign an (effective) temperature to the gas. Is the temperature you find here consistent with the one you determined in part (b)l

(Ill) Starting from the Maxwell distribution of speeds, Eq. 18-6, show (a) dv = N, and (b) J'OOv 2 f( v ) d v /N = 3kT/m.

(I) C 0 2 exists in what phase when the pressure is 30 atm and the temperature is 30C (Fig. 18-6)?

(I) (a) At atmospheric pressure, in what phases can C 0 2 exist? (b) For what range of pressures and temperatures can C 0 2 be a liquid? Refer to Fig. 18-6.

2 be a liquid? Refer to Fig. 18-6. 25. f n Water is in which nhase when the nressnre is 0.01 atm

(II) You have a sample of water and are able to control temperature and pressure arbitrarily, (a) Using Fig. 18-5, describe the phase changes you would see if you started at a temperature of 85C, a pressure of 180 atm, and decreased the pressure down to 0.004 atm while keeping the temperature fixed. (b) Repeat part (a) with the temperature at 0.0C. Assume that you held the system at the starting conditions long enough for the system to stabilize before making further changes.

(I) What is the partial pressure of water vapor at 30C if the humidity is 85%?

(I) What is the partial pressure of water on a day when the temperature is 25C and the relative humidity is 55%?

(I) What is the air pressure at a place where water boils at 80C?

(II) What is the dew point if the humidity is 75% on a day when the temperature is 25C?

(II) If the air pressure at a particular place in the mountains is 0.75 atm, estimate the temperature at which water boils.

(II) What is the mass of water in a closed room 5.0 m X 6.0 m X 2.4 m when the temperature is 24.0C and the relative humidity is 65%?

(II) What is the approximate pressure inside a pressure cooker if the water is boiling at a temperature of 120C? Assume no air escaped during the heating process, which started at 12C.

(II) If the humidity in a room of volume 440 m3 at 25C is 65%, what mass of water can still evaporate from an open pan?

(II) A pressure cooker is a sealed pot designed to cook food with the steam produced by boiling water somewhat above 100C. The pressure cooker in Fig. 18-17 uses a weight of mass m to allow steam to escape at a certain pressure through a small hole (diameter d) in the cookers lid. If d = 3.0 mm, what should m be in order to cook food at 120C? Assume that atmospheric pressure outside the cooker is 1.01 X 105 Pa.

(II) When using a mercury barometer (Section 13-6), the vapor pressure of mercury is usually assumed to be zero. At room temperature mercurys vapor pressure is about 0.0015 mm-Hg. At sea level, the height h of mercury in a barometer is about 760 mm. (a) If the vapor pressure of mercury is neglected, is the true atmospheric pressure greater or less than the value read from the barometer? (b) What is the percent error? (c) What is the percent error if you use a water barometer and ignore waters saturated vapor pressure at STP?

If the humidity is 45% at 30.0C, what is the dew point? Use linear internolation to find the temnerature nf the Hew

(Ill) Air that is at its dew point of 5C is drawn into a building where it is heated to 20 C. What will be the relative humidity at this temperature? Assume constant pressure of 1.0 atm. Take into account the expansion of the air.

(Ill) What is the mathematical relation between waters boiling temperature and atmospheric pressure? (a) Using the data from Table 18-2, in the temperature range from 50C to 150C, plot In P versus (1/7), where P is waters saturated vapor pressure (Pa) and T is temperature on the Kelvin scale. Show that a straight-line plot results and determine the slope and ^-intercept of the line, (b) Show that your result implies P = Be~AIT where A and B are constants. Use the slope and ^-intercept from your plot to show that A ~ 5000 K and B 7 X 1010 Pa.

(II) In the van der Waals equation of state, the constant b represents the amount of unavailable volume occupied by the molecules themselves. Thus V is replaced by (V nb), where n is the number of moles. For oxygen, b is about 3.2 X 10_5m3/mol. Estimate the diameter of an oxygen molecule.

(II) For oxygen gas, the van der Waals equation of state achieves its best fit for a = 0.14N-m4/mol2 and b = 3.2 X 10_5m3/mol. Determine the pressure in 1.0 mol of the gas at 0C if its volume is 0.70 L, calculated using (a) the van der Waals equation, (b) the ideal gas law.

(Ill) A 0.5-mol sample of 0 2 gas is in a large cylinder with a movable piston on one end so it can be compressed. The initial volume is large enough that there is not a significant difference between the pressure given by the ideal gas law and that given by the van der Waals equation. As the gas is slowly compressed at constant temperature (use 300 K), at what volume does the van der Waals equation give a pressure that is 5% different than the ideal gas law pressure? Let a = 0.14N*m4/mol2 and b = 3.2 X 10_5m3/mol.

(Ill) (a) From the van der Waals equation of state, show that the critical temperature and pressure are given by _ 8 a cr 27 bR a 21b2 [Hint: Use the fact that the P versus V curve has an inflection point at the critical point so that the first and second derivatives are zero.] (b) Determine a and b for C02 from the measured values of r cr = 304 K and Pcr = 72.8 atm.

How well does the ideal gas law describe the pressurized air in a scuba tank? (a) To fill a typical scuba tank, an air compressor intakes about 2300 L of air at 1.0 atm and compresses this gas into the tanks 12-L internal volume. If the filling process occurs at 20 C, show that a tank holds about 96 mol of air. (b) Assume the tank has 96 mol of air at 20C. Use the ideal gas law to predict the airs pressure within the tank, (c) Use the van der Waals equation of state to predict the airs pressure within the tank. For air, the van der Waals constants are a = 0.1373 N m4/mol2 and b = 3.72 X 10_5m3/mol. (d) Taking the van der Waals nressnre as the true air nressnre show that the ideal aas law

(II) At about what pressure would the mean free path of air molecules be (a) 0.10 m and (b) equal to the diameter of air molecules, 3 X 10-10m? Assume T = 20C.

(II) Below a certain threshold pressure, the air molecules (0.3-nm diameter) within a research vacuum chamber are in the collision-free regime, meaning that a particular air molecule is as likely to cross the container and collide first with the opposite wall, as it is to collide with another air molecule. Estimate the threshold pressure for a vacuum chamber of side 1.0 m at 20C.

(II) A very small amount of hydrogen gas is released into the air. If the air is at 1.0 atm and 15C, estimate the mean free path for a H2 molecule. What assumptions did you make?

(II) (a) The mean free path of C 0 2 molecules at STP is measured to be about 5.6 X 10-8 m. Estimate the diameter of a C 0 2 molecule, (b) Do the same for He gas for which m ~ 25 X 108 at STP

(II) (a) Show that the number of collisions a molecule makes per second, called the collision frequency, f , is given by / = v/u , and thus / = 4V2Trr2 vN /V . (b) What is the collision frequency for N2 molecules in air at T = 20 C and P = 1.0 X 10_2atm?

(II) We saw in Example 18-8 that the mean free path of air molecules at STP, M, is about 9 X 10_8m. Estimate the collision frequency / , the number of collisions per unit time.

(II) A cubic box 1.80 m on a side is evacuated so the pressure of air inside is 10_6torr. Estimate how many collisions molecules make with each other for each collision with a wall (0C).

(Ill) Estimate the maximum allowable pressure in a 32-cmlong cathode ray tube if 98% of all electrons must hit the screen without first striking an air molecule.

(I) Approximately how long would it take for the ammonia of Example 18-9 to be detected 1.0 m from the bottle after it is opened? What does this suggest about the relative importance of diffusion and convection for carrying odors?

(II) Estimate the time needed for a glycine molecule (see Table 18-3) to diffuse a distance of 15 jum in water at 20C if its concentration varies over that distance from 1.00mol/m3 to 0.50mol/m3? Compare this speed to its rms (thermal) speed. The molecular mass of glycine is about 75 u.

(II) Oxygen diffuses from the surface of insects to the interior through tiny tubes called tracheae. An average trachea is about 2 mm long and has cross-sectional area of 2 X 10-9 m2. Assuming the concentration of oxygen inside is half what it is outside in the atmosphere, (a) show that the concentration of oxygen in the air (assume 21% is oxygen) at 20C is about 8.7 mol/m3, then (b) calculate the diffusion rate /, and (c) estimate the average time for a molecule to diffuse in. Assume the diffusion constant is 1 X 10-5 m2/s.

A sample of ideal gas must contain at least N 106 molecules in order for the Maxwell distribution to be a valid description of the gas, and to assign it a meaningful temperature. For an ideal gas at STP, what is the smallest length scale (volume V = 3) for which a valid temperature can be assigned?

In outer space the density of matter is about one atom per cm3, mainly hydrogen atoms, and the temperature is about 2.7 K. Calculate the rms speed of these hydrogen atoms, and the pressure (in atmospheres).

Calculate approximately the total translational kinetic energy of all the molecules in an E. coli bacterium of mass 2.0 X 10-15 kg at 37C. Assume 70% of the cell, by weight, is water, and the other molecules have an average molecular mass on the order of 105 u.

(a) Estimate the rms speed of an amino acid, whose molecular mass is 89 u, in a living cell at 37C. (b) What would be the rms speed of a protein of molecular mass 85,000 u at 37C?

The escape speed from the Earth is 1.12 X 104m/s, so that a gas molecule travelling away from Earth near the outer boundary of the Earths atmosphere would, at this speed, be able to escape from the Earths gravitational field and be lost to the atmosphere. At what temperature is the average speed of (a) oxygen molecules, and (b) helium atoms equal tn 1 .1 9 .X 10 4 m /s? M Can vnu exn la in w h v ou r a tm o s-

The second postulate of kinetic theory is that the molecules are, on the average, far apart from one another. That is, their average separation is much greater than the diameter of each molecule. Is this assumption reasonable? To check, calculate the average distance between molecules of a gas at STP, and compare it to the diameter of a typical gas molecule, about 0.3 nm. If the molecules were the diameter of ping-pong balls, say 4 cm, how far away would the next ping-pong ball be on average?

A sample of liquid cesium is heated in an oven to 400 C and the resulting vapor is used to produce an atomic beam. The volume of the oven is 55 cm3, the vapor pressure of Cs at 400C is 17 mm-Hg, and the diameter of cesium atoms in the vapor is 0.33 nm. (a) Calculate the mean speed of cesium atoms in the vapor. (b) Determine the number of collisions a single Cs atom undergoes with other cesium atoms per second, (c) Determine the total number of collisions per second between all of the cesium atoms in the vapor. Note that a collision involves two Cs atoms and assume the ideal gas law holds.

Consider a container of oxygen gas at a temperature of 20C that is 1.00 m tall. Compare the gravitational potential energy of a molecule at the top of the container (assuming the potential energy is zero at the bottom) with the average kinetic enerav nf the molecules. Ts it reasonable to neglect

In humid climates, people constantly dehumidify their cellars to prevent rot and mildew. If the cellar in a house (kept at 20C) has 115 m2 of floor space and a ceiling height of 2.8 m, what is the mass of water that must be removed from it in order to drop the humidity from 95% to a more reasonable 40%?

Assuming a typical nitrogen or oxygen molecule is about 0.3 nm in diameter, what percent of the room you are sitting in is taken up by the volume of the molecules themselves?

A scuba tank has a volume of 3100 cm3. For very deep dives, the tank is filled with 50% (by volume) pure oxygen and 50% pure helium, (a) How many molecules are there of each type in the tank if it is filled at 20C to a gauge pressure of 12 atm? (b) What is the ratio of the average kinetic energies of the two types of molecule? (c) What is the ratio of the rms speeds of the two types of molecule?

A space vehicle returning from the Moon enters the atmosphere at a speed of about 42,000 km/h. Molecules (assume nitrogen) striking the nose of the vehicle with this speed correspond to what temperature? (Because of this high temperature, the nose of a space vehicle must be made of special materials; indeed, part of it does vaporize, and this is seen as a bright blaze upon reentry.)

At room temperature, it takes approximately 2.45 X 103 J to evaporate 1.00 g of water. Estimate the average speed of evaporating molecules. What multiple of vrms (at 20C) for water molecules is this? (Assume Eq. 18-4 holds.)

Calculate the total water vapor pressure in the air on the following two days: (a) a hot summer day, with the temperature 30C and the relative humidity at 65%; (b) a cold winter day, with the temperature 5C and the relative humidity at 75%.

At 300 K, an 8.50-mol sample of carbon dioxide occupies a volume of 0.220 m3. Calculate the gas pressure, first by assuming the ideal gas law, and then by using the van der Waals equation of state. (The values for a and b are given in Section 18-5.) In this range of pressure and volume, the van der Waals equation is very accurate. What percent error did you make in assuming ideal-gas-law behavior?

The density of atoms, mostly hydrogen, in interstellar space is about one per cubic centimeter. Estimate the mean free path of the hydrogen atoms, assuming an atomic diameter of 10-10 m.

Using the ideal gas law, find an expression for the mean free path that involves pressure and temperature instead of (N/V). Use this expression to find the mean free path for nitrogen molecules at a pressure of 7.5 atm and 300 K.

A sauna has 8.5 m3 of air volume, and the temperature is 90C. The air is perfectly dry. How much water (in kg) should be evaporated if we want to increase the relative humidity from 0% to 10%? (See Table 18-2.)

A 0.50-kg trash-can lid is suspended against gravity by tennis balls thrown vertically upward at it. How many tennis balls per second must rebound from the lid elastically, assuming they have a mass of 0.060 kg and are thrown at 12 m/s?

Sound waves in a gas can only propagate if the gas molecules collide with each other on the time scale of the sound waves period. Thus the highest possible frequency /max fr a sound wave in a gas is approximately equal to the inverse of the average collision time between molecules. Assume a gas, composed of molecules with mass m and radius r, is at pressure P and temperature T. (a) Show that /max - (b) Determine /max for 20C air at sea level. How many times larger is /max than the highest frequency in the human audio range (20 kHz)?

(II) Use a spreadsheet to calculate and graph the fraction of molecules in each 50-m/s speed interval from lOOm/s to 5000 m/s if T = 300 K.

(II) Use numerical integration [Section 2-9] to estimate (within 2%) the fraction of molecules in air at 1.00 atm and 20C that have a speed greater than 1.5 times the most probable speed.

(II) For oxygen gas the van der Waals constants are a = 0.14 N m4/mol2 and b = 3.2 X 10_5m3/mol. Using these values, graph six curves of pressure vs. volume between V = 2 X 10-5 m3 to 2.0 X 10-4 m3, for 1 mol of oxygen gas at temperatures of 80 K, 100 K, 120 K, 130 K, 150 K, and 170 K. From the graphs determine approximately the critical temperature for oxygen.