PV = nrt where R = universal gas constant

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Nelson Thomas
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1 Ideal Gas Law Dd Deduced dfrom Combination of Gas Relationships: V 1/P, Boyle's Law V, Charles's Law V n, Avogadro'sLaw Therefore, V nt/p or PV nt PV = nrt where R = universal gas constant The empirical Equation of State for an Ideal Gas

2 Ideal Gas Equation of State

3 Ideal Gas Law PV = nrt where R = universal gas constant R = PV/nT R = atm L mol 1 K 1 R = atm dm 3 mol 1 K 1 R = J mol 1 K 1 (SI unit) Standard molar volume = 22.4 L mol 1 at 0 C and 1 atm Real gases approach ideal gas behavior at low P & high T

4 ConcepTest #1 One mole of ice is similar in volume to A. The Lambert Glacier B. The iceberg that sank the Titanic C. The ice of the CU skating rink D. An ice cube in a cold drink E. A snowflake

5 ConcepTest #1 One mole of ice is similar in volume to A. The Lambert Glacier B. The iceberg that sank the Titanic C. The ice of the CU skating rink D. An ice cube in a cold drink E. A snowflake

6 ConcepTest #2 A steel vessel contains 1 mole of gas at 100K. 2 moles of gas are added and the temperature is increased to 200K. How does the pressure change? A. P increases by a factor of 4 B. P decreases by a factor of 4 C. P increases by a factor of 6 D. P decreases by a factor of 6 E. P does not change

7 ConcepTest #2 A steel vessel contains 1 mole of gas at 100K. 2 moles of gas are added and the temperature is increased to 200K. How does the pressure change? A. P increases by a factor of 4 B. P decreases by a factor of 4 C. P increases by a factor of 6 D. P decreases by a factor of 6 E. P does not change

8 ConcepTest #3 An ideal gas is compressed at constant temperature. Which picture below best describes how the volume V of the gas behaves as the pressure P is increased? A V B V P P C V D V P P

9 Dalton slaw Definition: Partial pressure P i is the pressure exerted by one component of a gas mixture (at total pressure P t ) P i = x i P t where x i is the mole fraction (x i = n i /n total ) Dalton s Law: P t = P 1 +P 2 +P P i The total pressure is equal to the sum of the partial pressures that each individual component gas would exert if it were alone

10 ConcepTest #4 A mixture of gases contains 4 g of He and 4 g of H 2. The total pressure is 300 Pa. What is the partial pressure of helium? A. 100 Pa B. 150 Pa C. 200 Pa D. 250 Pa

11 ConcepTest #4 A mixture of gases contains 4 g of He and 4 g of H 2. The total pressure is 300 Pa. What is the partial pressure of helium? A. 100 Pa B. 150 Pa C. 200 Pa D. 250 Pa

12 Graham s Law of Effusion Effusion is the movement of gases through small passages (e.g., passing through a plug of fine sand) where essentially all collisions are between gas and sand This feels like something where the faster you go, the faster you get through the maze, so we might expect an effusion rate velocity. Graham observed that for gases A and B Rate(A)/Rate(B) = [M(B)/M(A)] 1/2 The rate of effusion of a gas is inversely proportional to the square root of its molar mass (M)

13 Real Gases Compressibility, Z Z PV PV nrt RT Z = 1 Ideal Gas behavior Z < 1 PV less than expected Attractive forces Z > 1 PV greater than expected Repulsive forces Z Ar 1 Ideal Gas P

14 Real Gases data! Compressibility Z PV nrt

15 Boyle Temperature, T B Temperature of greatest extent of near-ideal behavior. We can determine T B analytically. Z T, P P df dt lim p 0 f T 0 at T T, the Boyle temperature t B

16 Real Gases CO 2 Look at 20 C isotherm. A B C Compression At C, liquid condensation begins D liquid- vapor mixture at P vap (20 C) E last vapor condenses F Steep rise in pressure A liquid or solid is much less compressible than a gas For T >T there is a single phase For T >T c, there is a single phase, with no liquid formed.

17 van der Waals equation of state Physically-motivated y y corrections to Ideal Gas EoS. For a real gas, both attractive and repulsive intermolecular forces are present. Empirical terms were developed to help account for both. 1. Repulsive forces: make pressure higher than ideal gas Excluded volume P nrt V nb Volume of one molecule of radius r is V m = 4/3 r 3 Closest approach of two molecules with radius r is 2r.

18 ConcepTest #5 nrt Excluded volume P V nb The volume of one molecule of radius r is V m = 4/3 r 3 The closest approach of two molecules l with radius r is 2r. What is the excluded volume for the two molecules? A. 2V m B. 4V m C. 8V m D. 16 V m

19 van der Waals equation of state Physically-motivated y y corrections to Ideal Gas EoS. For a real gas, both attractive and repulsive intermolecular forces are present. Empirical terms were developed to help account for both. 1. Repulsive forces: make pressure higher than ideal gas nrt Excluded volume P V nb Volume of one molecule of radius r is V m = 4/3 r 3 Closest approach of two molecules with radius r is 2r. The excluded volume V exc is 2 3 V m = 8V m for two molecules. b 4V N So we estimate that m A

20 van der Waals equation of state Physically-motivated y y corrections to Ideal Gas EoS. For a real gas, both attractive and repulsive intermolecular forces are present. Empirical terms were developed to help account for both. 2. Attractive forces: make pressure lower than ideal gas Pressure depends wall collisions, both on frequency and their force. Both scale as n/v, so we expect a pressure correction of the form a(n/v) 2, giving the van der Waals Equation of State 2 nrt an RT a P V nb V V b V 2 2

21 3D van der Waals eqn of state T= T/Tc

22 van der Waals Isotherms near T c v d W loops are not physical. Why? Patch up with Maxwell construction van der Waals Isotherms, T/T c

23 van der Waals Isotherms near T c Look at one of the van der Waals isotherms at a temperature re of 0.9 T c 1.5 G A D compress the gas at constant T, F G compress the liquid phase (steep and not very compressible) D F vapor condensing (gas and liquid coexist) These are stable states Reduced Pre essure, P r F C B g, l D T r = 0.9 A F C super cooled liquid D B superheated gas These are metastable t states tt Reduced d Volume, V r Metastable example: C B a non physical artifact of VdW Use a very clean glass. Add water and heat (patched up with Maxwell construction) for a while with a microwave oven (superheat) eat) Add a drop of sand or perhaps touch with a spoon.

24 Real Gases van der Waals EoS P an2 2 V nb nrt V Condensation Supercritical Fluid Solid Liquid fluid with T>T c, P>P c We can use vdw EoS to approximate features. E.g., P C V T C C a 2 27b 3 bn 8a 27bR P Triple point T SCF Critical point Gas T c P c

25 Critical Constants of Real Gases Notice that Z c is essentially 0.3 for all gases, while the other critical properties can be very different from each other. There is a lesson to learn here. We might think about looking at T and P in terms of reduced parameters: P r = P/P c and V r = V/V c

26 Critical Constants of Real Gases T r = T/T c

27 Next Steps... We have not invoked any molecular properties. It is as though the various gases were just different, structureless fluids. We will now change that!

van der Waals Isotherms near T c

van der Waals Isotherms near T c v d W loops are not physical. Why? Patch up with Maxwell construction van der Waals Isotherms, T/T c van der Waals Isotherms near T c Look at one of the van der Waals isotherms