van der Waals Isotherms near Tc

van der Waals Isotherms, T/Tc

v d W “loops” arenot physical. Why?

Patch up with Maxwellconstruction

van der Waals Isotherms near Tc

Look at one of the van der Waals isotherms at a temperature of 0.9 Tc

Reduced Volume, Vr

0.3 0.5 0.7 1 2 3 5 7 10

Red

uced

Pre

ssur

e, P

r

0.0

0.5

1.0

1.5

Tr = 0.9

D

B

G

F

C A

g, l

•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

F → C supercooled liquidD → B superheated gasThese are metastable states

C → B a non-physical artifact of vdW(patched up with Maxwell construction)

Metastable example:Use a very clean glass. Add water and heatfor a while with a microwave oven (superheat)Add a drop of sand or perhaps touch with a spoon.

Real Gases

Condensation Supercritical Fluidfluid with T>Tc, P>Pc

Solid

Gas

Liquid

P

T

SCF

Critical point

Triple point

P

an2

V2

V nb nRT

van der Waals EoS

Tc

Pc

We can use vdW EoS toapproximate features. E.g.,

2273

827

C

C

C

aPb

V bnaTbR

Critical Constants of Real Gases

Notice that Zc is essentially 0.3 for allgases, while the other critical propertiescan be very different from each other.

There is a lesson to learn here.

We might think about looking atT and P in terms of reduced parameters: Pr = P/Pc and Vr = V/Vc

Critical Constants of Real Gases

Tr = T/Tc

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!

Kinetic Theory of Gases

Everything up to this point has been empiricalThere is clearly a great deal of truth in the equationsHowever, there is no physical picture

To gain insight, one must develop and validate models We look at a molecular (kinetic theory of gases) modelto begin that process.We begin that process today

Reading Test #1

HCl and NH3 gases at the same temperature and pressure are introduced at opposite ends of a glass tube. A ring of solid NH4Cl forms wherethe two gases meet inside the tube.HCl + NH3 NH4ClWhere will the ring form?

A. At the centerB. Closer to the HCl endC. Closer to the NH3 end

Three Postulates ofKinetic Theory of Gases

3. Neither attractive nor repulsive forces exist between the particles except on contact (collision).

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.

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. HeB. H2

C. N2

D. NH3E. CH4

Kinetic-Molecular Theory

Consistent with Ideal Gas Law, Dalton’s Law,Graham’s Law and many other observations

Consider a moleculecolliding elastically with a wall

x-coordinate

velocity, -vx

velocity, vx

Wal

l

Relationship between Force and Change of Momentum

What is change of momentum on each collision?

Momentum, p = mvF = ma = m dv/dt = d(mv)dt = dp/dt

What is change of velocity on each collision?

v = vfinal – vinitial = vx – (– vx) = 2vx

p = 2vxm

Collision rate for one molecule with a wall

x (1ength of 1 side)

vx

-vx

Number of collisions in time t = vxt/2x

Wal

l

Box with dimensionsx by y by z

Number of collisions per unit time = vx /2x

Force (momentum change)per Unit Time for One Particle

dp/dt = Fx = (2vxm) (vx/2x)

= mvx2/x

This is the force exerted on the wall by one particle.

F= ma=dp/dt = (Change of Momentum per Collision) x

(Number of Collisions per Unit Time)

Pressure is Forceper Unit Area

Px = Fw/A = Fx/yz

Px = mvx2/xyz

Px = mvx2/V

x

yz

Fx = (2vxm) (vx/2x) = mvx2/x

What changes are required when N molecules and a distribution of velocities

are considered?

Multiply by N Replace vx

2 with

2 2xP m / V P Nm v /3Vxv

2vx

xy z

Px = mvx2/V

Px = Py = Pz = P implies2

2 2 2

3x y z

vv v v

and

V= xyz

Fundamental Equation from Simple Kinetic Theory of Gases

2PV Nm v /3

PV nRTParallels the ideal gas law:

2

2

Nm vnRT

33 3RTv

Mwith M (molar mass) = mN / n

nRTNm

2

rms3RT v vM

ConcepTest #3

At a given temperature, which of the following gases has the smallest vrms?

A. HeB. H2

C. N2

D. NH3E. CH4

ConcepTest #3

At a given temperature, which of the following gases has the smallest vrms?

A. HeB. H2

C. N2

D. NH3E. CH4

Connection between Kinetic Energy & Temperature

Average Kinetic Energy per Molecule

2m1 v2

Substitute this expression into the pressure equation to obtain the relationship between kinetic energy and T:

k

k B

3E RT (per mole)2

3or k T (per molecule)2

ConcepTest #4Flask A contains 8 g H2, and Flask B contains 8 g He. The flasks have the same volume and temperature. Compare the gas density (g/cm3), the kinetic energy per mole, and the total kinetic energy of the gases in the two flasks.

Density Ek /n Total KEA. A>B A>B A>BB. A>B A=B A>BC. A=B A=B A=BD. A=B A=B A>BE. A=B A>B A>B

ConcepTest #4Flask A contains 8 g H2, and Flask B contains 8 g He. The flasks have the same volume and temperature. Compare the gas density (g/cm3), the kinetic energy per mole, and the total kinetic energy of the gases in the two flasks.

Density Ek /n Total KEA. A>B A>B A>BB. A>B A=B A>BC. A=B A=B A=BD. A=B A=B A>BE. A=B A>B A>B