Lecture 24. Ideal Gas Law and Kinetic Theory

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1 Lecture 4 Ideal Gas Law and Kinetic Theory

2 Today s Topics: Ideal Gas Law Kinetic Theory of Gases Phase equilibria and phase diagrams

3 Ideal Gas Law An ideal gas is an idealized model for real gases that have sufficiently low densities. The condition of low density means that the molecules are so far apart that they do not interact except during collisions, which are effectively elastic. At constant volume the (absolute) pressure is proportional to the (absolute) temperature. P µ T

4 At constant temperature, the pressure is inversely proportional to the volume. P µ 1 V The pressure is also proportional to the amount of gas. n P µ n number of moles of gas

5 Moles and Avogadro s Number One mole of a substance contains as many particles as there are atoms in 1 grams of the isotope carbon-1. The number of atoms per mole is known as Avogadro s number, N A. N A mol -1 n N N A number of moles number of atoms

6 The mass per mole (in g/mol) of a substance has the same numerical value as the atomic or molecular mass of the substance (in atomic mass units). For example, hydrogen has an atomic mass of g/mol, while the mass of a single hydrogen atom is u. Mass of sample n m m particle particle N N A Mass m per mole Mass per mole

proportional to the volume of the gas. P nrt V n N N A R 8.

7 The absolute pressure of an ideal gas is directly proportional to the Kelvin temperature and the number of moles of the gas and is inversely proportional to the volume of the gas. P nrt V n N N A R 8.31J ( mol K) ideal gas constant ( mol K) k R 8.31J N A mol J K Boltzmann's constant

8 Consider a sample of an ideal gas that is taken from an initial to a final state, with the amount of the gas remaining constant. PV nrt PV T nr constant P V f T f f PV i T i i Constant T, constant n: Constant P, constant n: P fv f Pi Vi Boyle s law V f V i T T Charles law f i Constant V, constant n: P f T f P i T i Gay-Lussac s law

ACT: Ideal Gas Law I Two identical cylinders at the same temperature are filled with the same gas. If A contains three times as much gas as B, which cylinder has the higher pressure?

9 ACT: Ideal Gas Law I Two identical cylinders at the same temperature are filled with the same gas. If A contains three times as much gas as B, which cylinder has the higher pressure? a) cylinder A b) cylinder B c) both the same d) it depends on temperature T Ideal gas law: PV nrt Solve for pressure: P nr T V For constant V and T, the one with more gas (the larger value of n) has the higher pressure P.

10 ACT: Ideal Gas Law II Two cylinders at the same temperature are filled with the same gas. If B has twice the volume and half the number of moles as A, how does the pressure in B compare with the pressure in A? a) P B ½ P A b) P B P A c) P B ¼ P A d) P B 4 P A e) P B P A Ideal gas law: P nrt / V Because B has a factor of twice the volume, it has a factor of two less the pressure. But B also has half the amount of gas, so that is another factor of two reduction in pressure. Thus, B must have only one-quarter the pressure of A.

11 ACT: Balloon in Freezer What happens to the volume of a balloon if you put it in the freezer? a) it increases b) it does not change c) it decreases According to the Ideal Gas Law, when the temperature is reduced at constant pressure, the volume is reduced as well. The volume of the balloon therefore decreases. PV nrt DEMO: Balloon in liquid nitrogen

12 Example A sample of an ideal gas is originally at 0 C. What is the final temperature of the gas if both the pressure and volume are doubled? PV nrt PV T const

13 Example A sealed container has a volume of 0.00 m 3 and contains 15.0 g of molecular nitrogen (N ) which has a molecular mass of 8.0 u. The gas is at a temperature of 55 K. What is the absolute pressure of the nitrogen gas? PV P nrt nrt V n 15 g 8 g/mol mol

14 Kinetic Theory of Gases Gas molecules are in constant, random motion, colliding with each other and with the walls of the container. Each collision changes the particle s velocity. As a result, the atoms and molecules have different speeds.

15 THE DISTRIBUTION OF MOLECULAR SPEEDS v rms 970 m/s

16 Ideal Gases: Pressure, Temperature and Kinetic Energy The origin of pressure: å F x ma x m Dv Dt x D ( mv ) Dt x Average force Final momentum- Initial momentum Time between successivecollisions (- mv )- ( + mv ) x L v x x - mv L x

3 mean-square speed ø ö ç ç è æ ø ö ç è æ 3 3 L mv N L F A F P volume ø

17 L mv F x For our single molecule, the average force on the wall is: For N molecules in 3D, the average force is: ø ö ç ç è æ ø ö ç è æ L mv N F 3 mean-square speed ø ö ç ç è æ ø ö ç è æ 3 3 L mv N L F A F P volume ø ö ç ç è æ ø ö ç è æ V mv N P 3 ( ) ( ) rms mv rms N mv N PV NkT KE

18 THE INTERNAL ENERGY OF A MONATOMIC IDEAL GAS Average translational kinetic energy per particle: 1 mv 3 rms KE kt 3 U N kt 3 nrt Internal energy of an ideal gas

19 Example Assume that N behaves as an ideal gas and determine the rms speed of the nitrogen molecules when the temperature of the air is 93K. 1 3 mv rms kt m Mass per mol ( g / mol) -1 N ( mol ) A For nitrogen 3kT v rms m m 8.0g mol g mol 6 kg

20 Phase Equilibrium If a liquid is put into a sealed container so that there is a vacuum above it, some of the molecules in the liquid will vaporize. Once a sufficient number have done so, some will begin to condense back into the liquid. Equilibrium is reached when the numbers remain constant.

21 The pressure of the gas when it is in equilibrium with the liquid is called the equilibrium vapor pressure, and depends on the temperature. The vaporization curve determines the boiling point of a liquid: A liquid boils at the temperature at which its vapor pressure equals the external pressure.

22 This curve can be expanded. When the liquid reaches the critical point, there is no longer a distinction between liquid and gas; there is only a fluid phase. The fusion curve is the boundary between the solid and liquid phases; along that curve they exist in equilibrium with each other. The sublimation curve marks the boundary between the solid and gas phases. The triple point is where all three phases are in equilibrium. The phase diagram of water has a negative slope fusion curve.

23 The wonder of sweat! A liquid in a closed container will come to equilibrium with its vapor. However, an open liquid will not, as its vapor keeps escaping it will continue to vaporize without reaching equilibrium. As the molecules that escape from the liquid are the higher-energy ones, this has the effect of cooling the liquid. This is why sweating cools us off.

Lecture 24. Ideal Gas Law and Kinetic Theory

Lecture 24. Ideal Gas Law and Kinetic Theory Lecture 4 Ideal Gas Law and Kinetic Theory Today s Topics: Ideal Gas Law Kinetic Theory of Gases Phase equilibria and phase diagrams Ideal Gas Law An ideal gas is an idealized model for real gases that