Logarithm Review

ab = c, (a > 0, a ≠ 1) logac = b

Definition

If a = 10, it is called common logarithm

log c = log10c

If a = e = 2.718281828459045 ∙ ∙ ∙, it is called natural logarithm

ln c = logecKeys on your calculator

Properties of Logarithm

ln(xy) = ln(x) + ln(y)

ln(x/y) = ln(x) − ln(y)

ln(xm) = m ln(x)

Also see Appendix I B

x > 0, y > 0

Chapter 11

Liquids, Solids andIntermolecular Forces

continued

Vapor Pressure

Chemistry, continue on

Surface Molecules

Temperature: T Temperature: T

GA, 760 torr = 1 atm

H2O100 °C

NormalBoiling Point

Tibet, 480 torr < 1 atm

H2O85 °C

NormalBoiling Point

(a) The Vapor Pressure of Water, Ethanol, and Diethyl Ether as a Function of Temperature. (b) Plots of In(Pvap) versus 1/T for

Water, Ethanol, and Diethyl Ether

1/T (K−1)

T is in K!

Linear relation: y = kx + C

y

xC: intercept

slope: k = tg θ

θ

ln P = k(1/T) + C

Linear relation: y = kx + C

1/T (K−1)

Heat of vaporization ∆Hvap: energy needed to convert one mole

of liquid to gas. Unit: J/mol or kJ/mol.

∆Hvap > 0

slope k < 0

1ln vapHP C

R T

y x

ln (P)

1/T (K−1)

1

2

ln (P1)

ln (P2)

1/T1 1/T2

1ln vapHP C

R T

1

2 1 2

1 1ln vapHP

P R T T

Clausius-Clapeyron Equation

1ln vapHP C

R T

The vapor pressure of water at 25 °C is 23.8 torr, and the heat

of vaporization of water is 43.9 kJ/mol. Calculate the vapor

pressure of water at 50 °C.

Five: T1, T2, P1, P2, ∆Hvap

Four known, calculate the other.

1

2 1 2

1 1ln vapHP

P R T T

Clausius-ClapeyronequationR = 8.314 J · mol−1 · K−1

Units in ideal gas law

PV = nRT

P — atm, V — L, n — mol, T — K

Option 1

R = 0.082 atm · L · mol−1 · K−1

P — Pa, V — m3, n — mol, T — KOption 2

Chem 1211

Carbon tetrachloride, CCl4, has a vapor pressure of 213 torr at

40 °C and 836 torr at 80 °C. What is the normal boiling point of

CCl4?

1

2 1 2

1 1ln vapHP

P R T T

( Please try to work on this question by yourself. Will review next week)

Liquid potassium has a vapor pressure of 10.00 torr at 443 °C

and a vapor pressure of 400.0 torr at 708 °C. Use these data

to calculate

(a) The heat of vaporization of liquid potassium;

(b) The normal boiling point of potassium;

(c) The vapor pressure of liquid potassium at 100. °C.

( Please try to work on this question by yourself. Will review next week)

1

2 1 2

1 1ln vapHP

P R T T

Clausius-Clapeyron Equation

1ln vapHP C

R T

slope k < 0

1ln vapHP C

R T

y x

Linear relation: y = kx + C

y

xC: intercept

slope: k = tg θ

θ

a

b

c

d

Lines tilt to the right have positive slopes (a and b), left negative(c and d). Steeper line has greater absolute value of slope. In thisgraph, the order of slopes is

a > b > 0 > c > d

y

x

What is the order of heat of vaporization for these three substances?

Solids

Glass (SiO2)

Crystal

Noncrystal

Solid

Basis Crystal structure

The basis may be a single atom or molecule, or a small group of atoms, molecules, or ions.

NaCl: 1 Na+ ion and 1 Cl− ion

Cu: 1 Cu atom

Zn: 2 Zn atoms

Diamond: 2 C atoms

CO2: 4 CO2 molecules

=Use a point to represent the basis:

Lattice

Lattice point:

Unit cell: 2-D, at least a parallelogram

Unit cell is the building block of the crystal

How many kinds of 2-D unit cells

can we have?

Extend the concept of unit cell to 3-D,

the real crystals.

: 3-D, at least a parallelepiped

How many kinds of 3-D unit cells

can we have?

1. triclinic 2. monoclinic

3. orthorhombic

4. tetragonal5. rhombohedral (trigonal)

6. hexagonal7. cubic

The 14 Bravais lattices

7 crystal systems

a ≠ b ≠ cα ≠ β ≠ γ

a ≠ b ≠ c

α = β = γ = 90°

a = b ≠ cα = β = 90° ,γ = 120°

a = b = cα = β = γ = 90°

a = b ≠ cα = β = γ = 90°

a = b = c90° ≠ α = β = γ < 120°

γ

ab

ca

b

c

(Simple cubic)

Chem 1212: assume a lattice point is a single atom

• Size of the cell X-ray diffraction

Information of a cubic unit cell

The Wave Nature of LightThe Wave Nature of Light

• Number of atoms in a cell

• Size of the cell

• Size of the atoms Soon

X-ray diffraction

Now!

Information of a cubic unit cell

AB

C D

AB

C D E

F

Number of atoms in a unit cell = ¼ x 4 = 1

1 2 4

Number of Atoms in a Cubic Unit Cell

The body-centered cubic unit cell of a particular crystalline

form of iron is 0.28664 nm on each side. Calculate the density

(in g/cm3) of this form of iron. d = 7.8753 g/cm3

The body-centered cubic unit cell of a particular crystalline

form of an element is 0.28664 nm on each side. The density

of this element is 7.8753 g/cm3. Identify the element.

The face-centered cubic unit cell of a particular crystalline

form of platinum is 393 pm on each side. Calculate the

density (in g/cm3) of this form of platinum.

d = 21.4 g/cm3

Closest Packing

a a

aaa

a a

a a

aa

a a

a a a a a

a

a

b b b b

b b b b

b b b b

c c c c

c c c c

c c c c

· · · abab · · ·

· · · abab · · ·

1. triclinic 2. monoclinic

3. orthorhombic

4. tetragonal5. rhombohedral (trigonal)

6. hexagonal7. cubic

The 14 Bravais lattices

7 crystal systems

a ≠ b ≠ cα ≠ β ≠ γ

a ≠ b ≠ c

α = β = γ = 90°

a = b ≠ cα = β = 90° ,γ = 120°

a = b = cα = β = γ = 90°

a = b ≠ cα = β = γ = 90°

a = b = c90° ≠ α = β = γ < 120°

γ

· · · abcabc · · ·

abcabc = Cubic Closest Packing

e.g. Ag, 1 atoms (1 lattice point) in a unit cell

• Number of atoms in a cell

• Size of the cell

• Size of the atoms Soon

X-ray diffraction

Now!

Information of a unit cell

Now!

Example 11.7, page 494

Al crystallizes with a face-centered cubic unit cell. The radius of a Al atom is 143 pm. Calculate the density of solid Al in g/cm3.

r8L

L

r

2r

r

L

d = 2.71 g/cm3

What about simple cubic?

Simple Cubic

r

L

L = 2r

What about body-centeredcubic?

Body centered cubic

D

Body diagonal D = 4r

L

D

L

L F

L

Body diagonal D = 4r

r3

4L

L

Pythagorean theorem

Titanium metal has a body-centered cubic unit cell. Thedensity of titanium is 4.50 g/cm3. Calculate the edge lengthof the unit cell and a value for the atomic radius of titaniumin pm.

L = 328 pm

r = 142 pm

100-mL container

50 % 70 %

50 mL 70 mL

1 2 4

Packing Efficiency: fraction of volume occupied by atoms

74 %52 % 68 %

L = 2r r3

4L r8L

prove