chapter 1 part 2

1. borhan/cmt458/L2borhan/cmt458/L2 11
Units & Dimensional AnalysisUnits & Dimensional Analysis
LengthLength Meter (m)Meter (m)
MassMass Kilogram (kg)Kilogram (kg)
TimeTime Seconds (s)Seconds (s)
TemperatureTemperature Kelvin (K)Kelvin (K)
Amount ofAmount of
substancesubstance
Mole (mol)Mole (mol)
Electric CurrentElectric Current Ampere (A)Ampere (A)
LuminousLuminous
intensityintensity
Candela (cd)Candela (cd)
SI units (Base Units)

2. borhan/cmt458/L2borhan/cmt458/L2 22
Units & Dimensional AnalysisUnits & Dimensional Analysis
ForceForce Newton (N)Newton (N) 1 N = 1 kgms1 N = 1 kgms-2-2
PressurePressure Pascal(Pa)Pascal(Pa) 1 Pa = 1Nm1 Pa = 1Nm-2-2
EnergyEnergy Joule (J)Joule (J) 1 J = 1 Nm1 J = 1 Nm
Electric chargeElectric charge Coulomb(C)Coulomb(C) 1C=1 As1C=1 As
Electric potentialElectric potential
differencedifference
Volt (V)Volt (V) 1V=1 NmC1V=1 NmC-1-1
FrequencyFrequency HertzHertz 1Hz=1 s1Hz=1 s-1-1
SI Derived Units

3. borhan/cmt458/L2borhan/cmt458/L2 33
Conversion UnitsConversion Units
VolumeVolume L, mL,L, mL,
cmcm33
,dm,dm33
1 L= 1 dm1 L= 1 dm33
1 L=1000 cm1 L=1000 cm33
1 L=101 L=10-3-3
mm33
PressurePressure bar, atm, psi,bar, atm, psi,
torr, cm Hgtorr, cm Hg
1 atm = 760 torr1 atm = 760 torr
1 atm = 760 mm Hg1 atm = 760 mm Hg
1 atm = 101325 Pa1 atm = 101325 Pa
1 bar = 101 bar = 1055
PaPa
EnergyEnergy Cal, erg, eV,Cal, erg, eV,
cmcm-1-1
1 cal=4.184J1 cal=4.184J
1 erg = 101 erg = 10-7-7
JJ
1 eV=1.602x101 eV=1.602x10-19-19
JJ
1 cm1 cm-1-1
=1.987x10=1.987x10-23-23
JJ

4. borhan/cmt458/L2 4
Dimensional analysisDimensional analysis
Physical Quantity = numerical value x unitsPhysical Quantity = numerical value x units
If a quantity is dimensionless, it is just a number withoutIf a quantity is dimensionless, it is just a number without
unitsunits
Check for error in values used by checking dimensionalCheck for error in values used by checking dimensional
consistencyconsistency
Both sides of an equation have the same dimensionsBoth sides of an equation have the same dimensions
All terms of a sum have the same dimensionsAll terms of a sum have the same dimensions
Argument of a logarithm is dimensionlessArgument of a logarithm is dimensionless
Derivatives e.g dP/dT has the same dimension as P/TDerivatives e.g dP/dT has the same dimension as P/T
Integral of has dimensions of T2Integral of has dimensions of T2
∫TdT

5. borhan/cmt458/L2 5
Dimensional analysisDimensional analysis
Find the dimensions of the constants a andFind the dimensions of the constants a and
b in the van der Waals equationb in the van der Waals equation
nb has dimension of volumenb has dimension of volume
∴∴ bb →→ volume/ amountvolume/ amount
→→ pressurepressure
∴∴ aa →→ pressure x volume2pressure x volume2
amount2amount2
2
2
V
an
nbV
nRT
P −
−
=

6. borhan/cmt458/L2 6
Dimensional analysisDimensional analysis
Assume u = f(z, x)Assume u = f(z, x)
where A, B, C and D are parameterswhere A, B, C and D are parameters
Az has same units as BAz has same units as B22
CxCx22
has same units as Dhas same units as D33
ln ( ) and exp ( ) are dimensionless i.e unitlessln ( ) and exp ( ) are dimensionless i.e unitless
and their argument (D/E) and (E/D) are alsoand their argument (D/E) and (E/D) are also
dimensionless.dimensionless.
∴∴ E has same unit as DE has same unit as D












+
+
=
D
E
E
D
DCx
BAz
xzu expln),( 32
2

7. borhan/cmt458/L2 7
Dimensional analysisDimensional analysis
What is the unit of u(z,x)?What is the unit of u(z,x)?
Ans: Same asAns: Same as
If A, B, C, D, E, x and z are in SI units, thenIf A, B, C, D, E, x and z are in SI units, then
u(z,x) will also be SI unitsu(z,x) will also be SI units
Evaluate the following integralsEvaluate the following integrals and determineand determine
the resulting unitsthe resulting units
∫
2
1
T
T
dT ∫
2
1
2
P
P P
dP
∫
2
1
V
V
V
dV
3
2
D
B

8. borhan/cmt458/L2 8
Partial DerivativePartial Derivative
AA derivativederivative of aof a functionfunction that has more thanthat has more than
oneone independent variableindependent variable. Partial derivatives. Partial derivatives
are found by treating one independentare found by treating one independent
variable as avariable as a variablevariable and the rest asand the rest as
constantsconstants..