17-6 The Gas Laws and Absolute Temperature
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1 17-6 The Gas Laws and Absolute Temperature The relationship between the volume, pressure, temperature, and mass of a gas is called an equation of state. We will deal here with gases that are not too dense. Boyle s law: the volume of a given amount of gas is inversely proportional to the pressure as long as the temperature is constant.
2 17-6 The Gas Laws and Absolute Temperature The volume is linearly proportional to the temperature, as long as the temperature is somewhat above the condensation point and the pressure is constant. Extrapolating, the volume becomes zero at C; this temperature is called absolute zero.
3 17-6 The Gas Laws and Absolute Temperature The concept of absolute zero allows us to define a third temperature scale the absolute, or Kelvin, scale. This scale starts with 0Kat absolute zero, but otherwise is the same as the Celsius scale. Therefore, e e, the freezing point of water is K, and the boiling point is K. Finally, when the volume is constant, the pressure is directly proportional to the temperature.
4 17-6 The Gas Laws and Absolute Temperature Conceptual Example 17-9: Why you should not throw a closed glass jar into a campfire. What can happen if you did throw an empty glass jar, with the lid on tight, into a fire, and why?
5 17-7 The Ideal Gas Law We can combine the three relations just derived into a single relation: What about the amount of gas present? If the temperature and pressure are constant, the volume is proportional to the amount of gas:
6 17-7 The Ideal Gas Law A mole (mol) is defined as the number of grams of a substance that is numerically equal to the molecular mass of the substance: 1 mol H 2 has a mass of 2 g. 1 mol Ne has a mass of 20 g. 1 mol CO 2 has a mass of 44 g. The number of moles in a certain mass of material:
7 17-7 The Ideal Gas Law We can now write the ideal gas law: where n is the number of moles and R is the universal gas constant.
8 17-8 Problem Solving with the Ideal Gas Law Standard temperature and pressure (STP): T = 273 K (0 C) P = 1.00 atm = N/m 2 =1013kPa kpa. Example 17-10: Volume of one mole at STP. Determine the volume of 1.00 mol of any gas, assuming it behaves like an ideal gas, at tstp.
9 17-8 Problem Solving with the Ideal Gas Law Example 17-11: 11: Helium balloon. A helium party balloon, assumed to be a perfect sphere, has a radius of 18.0 cm. At room temperature (20 C), its internal pressure is 1.05 atm. Find the number of moles of helium in the balloon and the mass of helium needed to inflate the balloon to these values.
10 17-8 Problem Solving with the Ideal Gas Law Example 17-12: Mass of air in a room. Estimate the mass of air in a room whose dimensions are 5 m x 3 m x 2.5 m high, at STP.
11 17-8 Problem Solving with the Ideal Gas Law Volume of 1 mol of an ideal gas is 22.4 L If the amount of gas does not change: Always measure T in kelvins P must be the absolute pressure
12 Example 17-13: 13: Check tires cold Problem Solving with the Ideal Gas Law An automobile tire is filled to a gauge pressure of 200 kpa at 10 C. After a drive of 100 km, the temperature within the tire rises to 40 C. What is the pressure within the tire now?
13 17-9 Ideal Gas Law in Terms of Molecules: Avogadro s Number Since the gas constant is universal, the number of molecules in one mole is the same for all gases. That number is called Avogadro s number:
14 17-9 Ideal Gas Law in Terms of Molecules: Avogadro s Number Therefore we can write: or where k is called Boltzmann s constant.
15 17-9 Ideal Gas Law in Terms of Molecules: Avogadro s Number Example 17-14: Hydrogen atom mass. Use Avogadro s number to determine the mass of a hydrogen atom.
16 Example 17-15: How many molecules in one breath? Estimate how many molecules you breathe in with a 1.0-L breath of air.
17 18-1 The Ideal Gas Law and the Molecular Interpretation of Temperature Assumptions of kinetic theory: large number of molecules, moving in random directions with a variety of speeds molecules are far apart, on average molecules obey laws of classical mechanics and interact only when colliding collisions are perfectly elastic
18 18-1 The Ideal Gas Law and the Molecular Interpretation of Temperature The force exerted on the wall by the collision of one molecule is Then the force due to all molecules colliding with that wall is
19 18-1 The Ideal Gas Law and the Molecular Interpretation of Temperature The averages of the squares of the speeds in all three directions are equal: So the pressure is:
20 18-1 The Ideal Gas Law and the Molecular Interpretation of Temperature Rewriting, so The average translational kinetic energy of the molecules in an ideal gas is directly proportional to the temperature of the gas.
21 18-1 The Ideal Gas Law and the Molecular Interpretation of Temperature Example 18-1: Molecular kinetic energy. What is the average translational kinetic energy of molecules in an ideal gas at 37 C?
22 18-1 The Ideal Gas Law and the Molecular Interpretation of Temperature We can now calculate the average speed of molecules in a gas as a function of temperature:
23 18-1 The Ideal Gas Law and the Molecular Interpretation of Temperature Example 18-2: Speeds of air molecules. What is the rms speed of air molecules (O 2 and N 2 ) at room temperature (20 C)?
24 18-1 The Ideal Gas Law and the Molecular Interpretation of Temperature Conceptual Example 18-3: Less gas in the tank. A tank of helium is used to fill balloons. As each balloon is filled, the number of helium atoms remaining in the tank decreases. How does this affect the rms speed of molecules remaining in the tank?
25 18-1 The Ideal Gas Law and the Molecular Interpretation of Temperature Example 18-4: Average speed and rms speed. Eight particles have the following speeds, given in m/s: 1.0, 6.0, 4.0, 2.0, 6.0, 3.0, 2.0, 5.0. Calculate (a) the average speed and (b) the rms speed.
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