Problem 54
At the surface of Venus the average temperature is a balmy 460\(^\circ\)C due to the greenhouse effect (global warming!), the pressure is 92 earth-atmospheres, and the acceleration due to gravity is 0.894\(g{_e}{_a}{_r}{_t}{_h}\). The atmosphere is nearly all CO\(_2\) (molar mass 44.0 g/mol), and the temperature remains remarkably constant. Assume that the temperature does not change with altitude. (a) What is the atmospheric pressure 1.00 km above the surface of Venus? Express your answer in Venus-atmospheres and earth-atmospheres. (b) What is the root-mean-square speed of the CO\(_2\) molecules at the surface of Venus and at an altitude of 1.00 km?
Problem 55
An automobile tire has a volume of 0.0150 m\(^3\) on a cold day when the temperature of the air in the tire is 5.0\(^\circ\)C and atmospheric pressure is 1.02 atm. Under these conditions the gauge pressure is measured to be 1.70 atm (about 25 lb/in.\(^2\)). After the car is driven on the highway for 30 min, the temperature of the air in the tires has risen to 45.0\(^\circ\)C and the volume has risen to 0.0159 m\(^3\). What then is the gauge pressure?
Problem 56
A flask with a volume of 1.50 \(L\), provided with a stopcock, contains ethane gas (C\(_2\)H\(_6\)) at 300 K and atmospheric pressure (1.013 \(\times\) 10\(^5\) Pa). The molar mass of ethane is 30.1 g/mol. The system is warmed to a temperature of 550 \(K\), with the stopcock open to the atmosphere. The stopcock is then closed, and the flask is cooled to its original temperature. (a) What is the final pressure of the ethane in the flask? (b) How many grams of ethane remain in the flask?
Problem 57
A balloon of volume 750 m\(^3\) is to be filled with hydrogen at atmospheric pressure (1.01 \(\times\) 10\(^5\) Pa). (a) If the hydrogen is stored in cylinders with volumes of 1.90 m\(^3\) at a gauge pressure of 1.20 \(\times\) 10\(^6\) Pa, how many cylinders are required? Assume that the temperature of the hydrogen remains constant. (b) What is the total weight (in addition to the weight of the gas) that can be supported by the balloon if both the gas in the balloon and the surrounding air are at 15.0\(^\circ\)C? The molar mass of hydrogen (H\(_2\)) is 2.02 g/mol. The density of air at 15.0\(^\circ\)C and atmospheric pressure is 1.23 kg/m\(^3\). See Chapter 12 for a discussion of buoyancy. (c) What weight could be supported if the balloon were filled with helium (molar mass 4.00 g/mol) instead of hydrogen, again at 15.0\(^\circ\)C?
Problem 58
A vertical cylindrical tank contains 1.80 mol of an ideal gas under a pressure of 0.300 atm at 20.0\(^\circ\)C. The round part of the tank has a radius of 10.0 cm, and the gas is supporting a piston that can move up and down in the cylinder without friction. There is a vacuum above the piston. (a) What is the mass of this piston? (b) How tall is the column of gas that is supporting the piston?
Problem 59
A large tank of water has a hose connected to it (Fig. P18.59). The tank is sealed at the top and has compressed air between the water surface and the top. When the water height \(h\) has the value 3.50 m, the absolute pressure \(p\) of the compressed air is 4.20 \(\times\) 10\(^5\) Pa. Assume that the air above the water expands at constant temperature, and take the atmospheric pressure to be 1.00 \(\times\) 10\(^5\) Pa. (a) What is the speed with which water flows out of the hose when \(h\) = 3.50 m? (b) As water flows out of the tank, \(h\) decreases. Calculate the speed of flow for \(h\) = 3.00 m and for \(h\) = 2.00 m. (c) At what value of h does the flow stop?
Problem 60
A light, plastic sphere with mass \(m\) = 9.00 g and density \(\rho\) = 4.00 kg/m\(^3\) is suspended in air by thread of negligible mass. (a) What is the tension \(T\) in the thread if the air is at 5.00\(^\circ\)C and \(p\) = 1.00 atm? The molar mass of air is 28.8 g/mol. (b) How much does the tension in the thread change if the temperature of the gas is increased to 35.0\(^\circ\)C? Ignore the change in volume of the plastic sphere when the temperature is changed.
Problem 61
Estimate the number of atoms in the body of a 50-kg physics student. Note that the human body is mostly water, which has molar mass 18.0 g/mol, and that each water molecule contains three atoms.
Problem 62
A person at rest inhales 0.50 L of air with each breath at a pressure of 1.00 atm and a temperature of 20.0\(^\circ\)C. The inhaled air is 21.0% oxygen. (a) How many oxygen molecules does this person inhale with each breath? (b) Suppose this person is now resting at an elevation of 2000 m but the temperature is still 20.0\(^\circ\)C. Assuming that the oxygen percentage and volume per inhalation are the same as stated above, how many oxygen molecules does this person now inhale with each breath? (c) Given that the body still requires the same number of oxygen molecules per second as at sea level to maintain its functions, explain why some people report 'shortness of breath' at high elevations.
Problem 63
You have two identical containers, one containing gas \(A\) and the other gas \(B\). The masses of these molecules are \(m$$_A\) = 3.34 \(\times\) 10\({^-}{^2}{^7}\) kg and \(m$$_B\) = 5.34 \(\times\) 10\({^-}{^2}{^6}\) kg. Both gases are under the same pressure and are at 10.0\(^\circ\)C. (a) Which molecules (\(A\) or \(B\)) have greater translational kinetic energy per molecule and rms speeds? (b) Now you want to raise the temperature of only one of these containers so that both gases will have the same rms speed. For which gas should you raise the temperature? (c) At what temperature will you accomplish your goal? (d) Once you have accomplished your goal, which molecules (\(A\) or \(B\)) now have greater average translational kinetic energy per molecule?
