1
Differential Scanning Calorimetry
sample
reference
Linear Temperature Scan
Tsample Tref
T
dT
= 20C min 1
dt
+
Heater
sample
power
monitor
Heat flow =
Scan
Control
Heater
T
reference
power
monitor
qp
dt
Jq =
time
endotherm
+
dq
dt
trCp
heat flux
mJ s1
e
Partial Derivative Conversion
= CV +
V
=
U
P
+ V
T V TV
=
1
1
=
Cp
H
T P
H = U + PV
definition
invert
misplaced numerator
misplaced numerator
H
T V
T
HP
H
VT
misplaced denominator
H
T V
T
PH
misplaced constant variable
chain rule
misplaced consta
Application of the First Law to Ideal Gases
Calculate q,w, U, and H for ideal gas processes:
dU = q + w
dU = Cv dT dH = Cp dT
U
U = q + w
for any process since = 0
VT
Isothermal Reversible Expansion
dT = 0 so dU =
and dH =
q = w
V2
V1
w = nRT ln
P2V2 = P
LINEST in Excel The Excel spreadsheet function "linest" is a complete linear least squares curve fitting routine that produces uncertainty estimates for the fit values. There are two ways to access the "linest" functionality; through the function directly
Integrating the Basic Derivatives
dV = V dT
cst. P
Good: V Vo
V
V
T
dV =
Vo
V dT
To
V
o
VVo = Vo (T  To)
T
V = Vo T
To
Better: V Vo + Vo (T  To)
V
V
T
dV =
Vo
(Vo + Vo (T  To) dT
To
V
o
VVo = V o (T  To) + Vo 2
(T  To)2
2
T
To
Best: dV = dT
V
V
Error Analysis
Significant Figures in Calculations
Every lab report must have an error analysis. For many experiments, significant
figure rules are sufficient. For a brush up on significant figure rules, see your General
Chemistry or Analytical text. Reme
Van der Waals Liquifaction
90
(atm)
80
70
vapor
liquid
2phase
60
P
P
(atm)
supercritical fluid
40
30
50
20
V (L)
0
0.2
0.4
0.6
an2
P + V2 (V  nb) = nRT
2P
I. ( 2)T = 0
V
II. (
RTc a
Pc= V b  2
c
Vc
R
P
)T = (VcTc 2 + 2a =
b) V3
V
c
III. (
2
2P
)T = (
Work
dU = dq + dw
dU = dq PextdV
dw = F dx
dU = dq PextdV + F dx
(mN/m)
72.8
22.3
18.4
51.1
P, T
dq
Surface Tension: dw = d
Interface
water/air
ethanol/air
hexane/air
hexane/water
dV
Tsurr
J/m2 or N/m
dr
d = 8r dr
From r to r + dr
dU = dq PextdV + d
Ext
Equipartition Theorem
Goal Predict U and Cv for gases
translation
rotation
vibration
z
y
x
kinetic only
p2
2m
kinetic only
J2
2I
kinetic and potential
p2 1
2
2m + 2 kx
U
monatomic gas experimental U = 3/2 RT or Cv = ( T )V = 3/2R
1/2 R for each quadratic
Taylor Series
f(x) = f(xo) + f'(xo)
(xxo)
(xxo)2
(xxo)3
(xxo)n
+ f"(xo) 2! + f"'(xo) 3! + . + f(n)(xo) n!
1!
order n
functional form of fit
0
constant
y= c
f(x)
xo
xo
x
xo
straight line
x
xo
1
x
x
y = ax + b
f(x)
2
quadratic
y = ax2+bx + c
f(x)
3
cubi
JouleThomson Expansion
insulated q=0
n P1
n P2
V1 T1
V2 T 2
P High
P Low
adiabatic expansion: q=0
work:
left: P1V1
w = P1V1 P2V2
U = U2  U1 = q + w = w
U2  U1 = P1V1 P2V2
U2 + P2V2 = U1 + P1V1
H2 = H1
JT = (
T
)
P H
0 = dH = (
0=(
H
) dP + (H) dT
P T
T
Temperature Dependence of Reacton Enthalpy, rH
rHT1
T1:
R
re
Cp
(T1 T2)

rHT2
R
T2:
P
it doesnt matter how you get there!

pr
Cp
re
(T2 T1)
P
pr
rHT2 = Cp (T1 T2) + rHT1 + Cp (T2 T1)
pr
re
rHT2 = rHT1 + (Cp  Cp ) (T2 T1)
pr
re
rHT2 rHT1 = rCp T
rCp = C
Basic Derivatives
V(T,P)
V
V
dV = ( T )P dT + ( P )T dP
1 V
= V ( )P
T
isobaric thermal expansivity, coefficient of thermal expansion
1 V
= = V ( )T
P
isothermal compressibility
dV = V dT  V dP
do and give everything we need to know about mechanical be
Cp vs. Cv
H = U + PV
U
U
Cv = T
V
H
Cp = T
P
H
H
U
T
H
(U + PV)
U
V
= +P
I. Cp = T =
T
P
P T P
T P
U
V
U
II. Cp Cv = T + P T T
P
P V
U
U
III. dU = T dT + V dV
V
T
U
U dT U dV
IV. T = T dT + V dT
P V
T
at cst. P
U
U
U V
V. T = T + V T
P V