van der Waals Isotherms near T c

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Karin Murphy
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1 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

2 van der Waals Isotherms near T c Look at one of the van der Waals isotherms at a temperature 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 Pressure, P r F C B g, l D T r = 0.9 A F C supercooled liquid D B superheated gas These are metastable states C B a non-physical artifact of vdw (patched up with Maxwell construction) Reduced Volume, V r Metastable example: Use a very clean glass. Add water and heat for a while with a microwave oven (superheat) Add a drop of sand or perhaps touch with a spoon.

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

different from each other. There is a lesson to learn here.

4 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

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

6 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!

7 Kinetic Theory of Gases Everything up to this point has been empirical There is clearly a great deal of truth in the equations However, there is no physical picture To gain insight, one must develop and validate models We look at a molecular (kinetic theory of gases) model to begin that process. We begin that process today

8 Reading Test #1 HCl and NH 3 gases at the same temperature and pressure are introduced at opposite ends of a glass tube. A ring of solid NH 4 Cl forms where the two gases meet inside the tube. HCl + NH 3 NH 4 Cl Where will the ring form? A. At the center B. Closer to the HCl end C. Closer to the NH 3 end

9 Three Postulates of Kinetic Theory of Gases 1. Gas composed of hard spherical particles that are small relative to the mean distances between them. 2. Particles are in constant motion and have kinetic energy. 3. Neither attractive nor repulsive forces exist between the particles except on contact (collision).

10 ConcepTest #2 Which of the following gases deviates most from the postulates of kinetic molecular theory? (i.e., deviates most from ideal gas behavior) A. He B. H 2 C. N 2 D. NH 3 E. CH 4

11 Kinetic-Molecular Theory Consistent with Ideal Gas Law, Dalton s Law, Graham s Law and many other observations

12 Consider a molecule colliding elastically with a wall velocity, v x Wall x-coordinate velocity, -v x

13 Relationship between Force and Change of Momentum Momentum, p = mv F = ma = m dv/dt = d(mv)dt = dp/dt What is change of velocity on each collision? v= v final v initial = v x ( v x )= 2v x What is change of momentum on each collision? p = 2v x m

14 Collision rate for one molecule with a wall Wall v x -v x Box with dimensions x by y by z x (1ength of 1 side) Number of collisions in time t = v x t/2x Number of collisions per unit time = v x /2x

15 Force (momentum change) per Unit Time for One Particle F= ma=dp/dt = (Change of Momentum per Collision) x (Number of Collisions per Unit Time) dp/dt = F x = (2v x m) (v x /2x) = mv x2 /x This is the force exerted on the wall by one particle.

16 Pressure is Force per Unit Area y z x P x = F w /A = F x /yz F x = (2v x m) (v x /2x) = mv x2 /x P x = mv x2 /xyz P x = mv x2 /V

17 What changes are required when N molecules and a distribution of velocities are considered? Multiply by N Replace v x2 with 2 v x P x = P y = P z = P implies and v v v x y z v 2 3 P x = mv x2 /V 2 2 P m v /V P Nm v /3V x x y z x V= xyz

18 Fundamental Equation from Simple Kinetic Theory of Gases 2 PV Nm v /3 Parallels the ideal gas law: PV nrt 2 Nm v nrt 3 2 3nRT 3RT v v Nm M with M (molar mass) = mn / n 2 v rms 3RT M

19 ConcepTest #3 At a given temperature, which of the following gases has the smallest v rms? A. He B. H 2 C. N 2 D. NH 3 E. CH 4

20 ConcepTest #3 At a given temperature, which of the following gases has the smallest v rms? A. He B. H 2 C. N 2 D. NH 3 E. CH 4

21 Connection between Kinetic Energy & Temperature Average Kinetic Energy per Molecule 1 2 Substitute this expression into the pressure equation to obtain the relationship between kinetic energy and T: m 3 E RT (per mole) k 2 3 or k T (per molecule) k B 2 v 2

22 ConcepTest #4 Flask A contains 8 g H 2, and Flask B contains 8 g He. The flasks have the same volume and temperature. Compare the gas density (g/cm 3 ), the kinetic energy per mole, and the total kinetic energy of the gases in the two flasks. Density E k /n Total KE A. A>B A>B A>B B. A>B A=B A>B C. A=B A=B A=B D. A=B A=B A>B E. A=B A>B A>B

23 ConcepTest #4 Flask A contains 8 g H 2, and Flask B contains 8 g He. The flasks have the same volume and temperature. Compare the gas density (g/cm 3 ), the kinetic energy per mole, and the total kinetic energy of the gases in the two flasks. Density E k /n Total KE A. A>B A>B A>B B. A>B A=B A>B C. A=B A=B A=B D. A=B A=B A>B E. A=B A>B A>B

PV = nrt where R = universal gas constant

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