Formulae Sheet

Fundamental constants

R = 0.08314 dm3 bar K-1 mol-1; R = 0.08206 dm3 atm K-1 mol-1; R = 8.314 J K-1 mol-1

Boltzmann constant kB 1.381 x 10–23 J K–1 Avogadro constant NA 6.022 x 1023 mol–1

Gas Laws

Boyle’s law: PV = constant, at constant n, T

Avogadro’s principle: V = constant x n, at constant P, T

Charles’s law: V = constant x T, at constant n, P or P = constant x T, at constant n, V

Density of a gas: ρ = M × P/ RT

Dalton’s law of partial pressures : Ptotal = P1 + P2 + P3 + ...

n1
Partial pressure P1 = x1 . Ptotal Mole fraction x1 =
n total
Pi M g
Barometric distribution law: ln  i h
Pi ,0 RT
Molar kinetic energy : 3
E RT
2
Molecular Speed

2 RT 8 RT
Most probable speed u*  Mean speed u 
M M

3RT
Root mean square speed u rms 
M
1 1
rate of effusion  rate of diffusion 
M M

PA PAN A
rate of effusion (molecules/s)  
 2mk BT  1/ 2
 2MRT  1 / 2

u 1
Molecular collision rate: Zmol  2[X]ud 2 mean free path:   Z 
mol 2[X ]d 2

i
Equation of State

Virial equation of state: PVm  RT (1  B' P  C' P 2  ...)

B C
PVm  RT (1   2  ...)
Vm Vm

2
nRT n RT a
van der Waals equation of state P  a  P  2
V  nb V Vm  b Vm

a 8a
Critical constants Vc  3b Pc  Tc 
27 b 2 27Rb

Reduced variables P Vm T
Pr  Vr  Tr 
Pc Vc Tc

0 Vm PVm
Boyle temperature pVm  RTB (1   ...) Compression factor Z  
Vm Vmo RT
pVm  RTB

First law of thermodynamics U  q  w

Work

Definition w   F  dl     Pexternal dAdl    Pexternal dV

Vf
Reversible isothermal expansion dV V
w  nRT   nRT ln f
Vi
V Vi

Heat capacities
 U 
Heat capacity at constant volume Cv    qv = CvΔT
 T  V

Heat capacity at constant pressure  H 

Cp   
 T  P

C p  C v  nR C p, m  C v ,m  R
Tf Tf

U(Tf , Vf )  U(Ti , Vi )  U   C v dT  n  C v , m dT
Ti Ti

ii
Debye extrapolation: Cp = aT3

Enthalpy

Enthalpy H = U + PV ΔH = qp = CpΔT

Reaction enthalpy rHo  

products
v f H o   v
reac tan ts
f Ho

T2
Temperature dependence of reaction enthalpy  r H o (T2 )   r H o (T1 )   C
o
r p dT
T1

where  r C op   vC
products
o
p,m   vC
reac tan ts
o
p,m

1 

Adiabatic changes Tf  Vf 
Pi Vi  Pf Vf   where   C p,m /C v, m
Ti  Vi 
Calorimetry q = c x mass x DT

Internal pressure  U 
T   
 V  T

1  V 
T    
Expansion coefficient 1  V  Isothermal compressibility V  P  T
  
V  T  P

 2 TV
C P C V 
kT

Joule-Thomson coefficient m = (δT/δP)H

Isothermal Joule-Thomson coefficient mT = (δH/δP)T

mT = -CPm
dH  C p dP  C p dT

dU   T dV  C V dT

Entropy
dq rev
S = k ln W dS 
T

iii
Second law of thermodynamics Suniverse = Ssystem + Ssurroundings > 0

Clausius inequality dS >dq/T

Vf
For reversible isothermal changes, ΔS  nR ln
Vi
Pf
ΔS   nR ln
Pi

For reversible change in temperature at constant volume, dq reversible  C V dT

nC V,m dT T
ΔS    nC V,m ln f
T Ti

For a reversible change in temperature at constant P, dq reversible  C P dT

nC P ,m dT Tf
S    nC P , m ln
T Ti

For a change ViTi → VfTf, Vf T

ΔS  nR ln  nC V,m ln f
Vi Ti

Pf T
For a change PiTi → PfTf, ΔS   nR ln  nC P,m ln f
Pi Ti

Liquid to gas dq reversible q H vaporization

ΔSvaporization    reversible 
T Tvaporization Tvaporization

dq reversible q reversible H fusion

Solid to liquid ΔSfusion    
T Tfusion Tfusion

H fusion b C p (liq)dT H vap

Tf T T
Absolute entropy C p (s)dT C (g )dT
S(T )  S(0)       p
0
T Tf Tf
T Tb Tb
T

Standard reaction entropy Sor   vS

products
m   vS
reac tan ts
m

iv
Efficiency

q cold T  w cycle Tcold

  cold   1 
q hot Thot q hot Thot

Tcold q
Maximum performance factor for refrigerator = 
Thot  Tcold w

Helmholtz energy A = U – TS

Gibbs energy

G = H – TS

Standard reaction Gibbs energy DrGO = ΔrHo – TΔrSo

r G   v f G o   v f G o
products reac tan ts

dU = TdS – PdV dG = VdP – SdT

Gibbs-Helmholtz equation   ( G / T )  H G (T2 ) G (T1)  1 1 

   2   H(T1 )  
 T P T T2 T1  T2 T1 

Pf P
nRT f
dP P
For ideal gases, G   dP  nRT   nRT ln f
Pi
P Pi
P Pi

 G 
Chemical potential i   
 ni T , P , n j ( j i )

Fundamental equation dG  VdP -SdT   μi dni

i

For ideal gases,  m   mo  RT ln( P / bar )

f
Fugacity, f  P  fugacity coefficient For real gases,    o  RTln
Po
ni  ni ,o  G 
Extent of reaction  Gibbs reaction energy,  r G   
i   T , P

 r G   r G o  RT ln Q Δr G o   RT ln K Po
v
K p  K c (RT ) 

K2 Δr H o 1 1
van’t Hoff equation ln    
K1 R  T2 T1 

Rate laws

1 1  [ B ] /[ B ]o 
First order : [ A]t  [ A]o e
 kt
Second order:   kt ln   ([ B] o [ A]o )kt
[ A] [ A]o  [ A] /[ A]o 

Zero order: [A]t = -kt + [A]o

Half-life

ln 2 0.693 1 [ A]o
First-order: t1/ 2   Second-order: t1 / 2  Zero-order: t1/ 2 
k k k[ A]o 2k
Arrhenius equation:  Ea RT
k  Ae
Enzyme kinetics: k 2 [ E ]o 1 1 K  1
v    M 
1  K M /[ S ] v vmax  Vmax  [S ]

dP ΔS m ΔH m Tf
Clapeyron equation:  P  ln
dT ΔVm Vm Ti

Clausius-Clapeyron equation:
Pf H vap  1 1
ln    
Pi R  T f Ti 

Raoult’s law: Psolvent = xsolventPosolvent

Henry’s law: PB = bBKB

Colligative properties:

Boiling point elevation: DT = Kbb

Freezing point depression : DT = Kf b

van’t Hoff equation: ΠVsoln = nsoluteRT

vi