van der Waals Isotherms near Tc - JILAwcl/Chem4511/images/Gases 4 4511 fall 2… · van der Waals...
Transcript of van der Waals Isotherms near Tc - JILAwcl/Chem4511/images/Gases 4 4511 fall 2… · van der Waals...
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