Formula sheet_Final

8/7/2019 Formula sheet_Final

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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 k B 1.381 x 10 –23 J K –1 Avogadro constant N A 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 + ...

Partial pressure P1 = x1 . Ptotal Mole fraction x1 =total

1

n

n

Barometric distribution law:

Molar kinetic energy :

Molecular Speed

Most probable speedM

RT *u

2= Mean speed

M

RT u

π

8=

Root mean square speedM

RT urms

3=

Molecular collision rate: mean free path:

i

M

1 effusionof rate α

M

1 diffusionof rate α

h RT

g M

P

P ln i

,i

i ⋅⋅

−=0

RT E 2

3=

( ) ( ) 2/1

A

2/1B MRT2

PAN

Tmk 2

PA/s)(moleculeseffusionof rate

π

=

π

=

2mol du]X[2Z π= 2

mol d]X[2

1

Z

u

π==λ



Equation of State

Virial equation of state:

van der Waals equation of state

Critical constants

Reduced variables

Boyle temperature

First law of thermodynamics

Work

Definition

Reversible isothermal expansion

Heat capacities

Heat capacity at constant volume qv = CvΔT

Heat capacity at constant pressure

ii

...)V

C

V

B1(RTPV

...)P'CP'B1(RTPV

2

mm

m

2

m

+++=

+++=

Bm

m

Bm

RT pV

...)V

01(RT pV

≈

++=

2

mm

2

V

a

bV

RTP

V

na

nbV

nRTP −

−=

−

−=

Rb27

a8 T

b27

a P b3V c2cc ===

c

r

c

mr

c

r T

TT

V

VV

P

PP ===

wqU +=∆

∫ ∫ ∫ ∫ −=−=⋅= dV P dAdl P dl F w external external

i

f

V

VV

VlnnRT

V

dVnRTw

f

i

−=−= ∫

V

vT

UC

∂

∂=

P

pT

HC

∂∂

=

R CC nR CC m,vm, pv p =−=−

RT

PV

V

VZfactor nCompressio m

o

m

m ==

∫ ∫ ==∆=−f

i

f

i

T

T

m,v

T

T

viif f dTCndTCU)V,T(U)V,T(U



Debye extrapolation: C p = aT3

Enthalpy

Enthalpy H = U + PV ΔH = q p = C p ΔT

Reaction enthalpy

Temperature dependence of reaction enthalpy

Adiabatic changes

Calorimetry q = c x mass x ∆ T

Internal pressure

Expansion coefficient Isothermal compressibility

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

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

µ T = -C P µ

Entropy

S = k ln W

iii

mv,m p,

1

i

f

i

f /CCwhereV

V

T

T=γ

=

γ −γ γ = f f ii VPVP

∑∑

∫ −=∆

∆+∆=∆

tstanreac

o

m, p

products

o

m, p

o

pr

T

T

o

pr 1

o

r 2

o

r

vCvCCwhere

dTC)T(H)T(H2

1

T

TV

U

∂∂=π

dTCdVdU VT +π=

T

V

V

1

P

∂∂

=α

T

T

P

V

V

1

∂∂

−=κ

T

2

VPk TVCC α=−

dTCdPCdH p p

+µ−=

T

dqdS rev=

∑ ∑ ∆−∆=∆ products tstanreac

o

f

o

f

o

r HvHvH



Second law of thermodynamics ∆ Suniverse = ∆ Ssystem + ∆ Ssurroundings > 0

Clausius inequality dS >dq/T

For reversible isothermal changes,

For reversible change in temperature at constant volume,

For a reversible change in temperature at constant P,

For a change V iT i → V f T f ,

For a change P iT i → P f T f ,

Liquid to gas

Solid to liquid

Absolute entropy

Standard reaction entropy

iv

i

f

i

f

P

PlnnR ΔS

V

VlnnR ΔS

−=

=

i

f V,m

V,m

Vreversible

T

TlnnC

T

dTnCΔS

dTCdq

≈=

=

∫

i

f m,P

m,P

Preversible

TTlnnC

TdTnCS

dTCdq

≈=∆

=

∫

i

f V,m

i

f

T

TlnnC

V

VlnnR ΔS +=

i

f P,m

i

f

T

TlnnC

P

PlnnR ΔS +−=

onvaporizati

onvaporizati

onvaporizati

reversiblereversibleonvaporizati

T

H

T

q

T

dqΔS

∆=== ∫

fusion

fusion

fusion

reversiblereversiblefusion

T

H

T

q

T

dqΔS

∆=== ∫

∫ ∫ ∫ +∆

++∆

++=T

T

p

b

vap

T

T

p

f

fusion

T

0

p

b

b

f

f

T

dT)g(C

T

H

T

dT)liq(C

T

H

T

dT)s(C)0(S)T(S

∑∑ −=∆tstanreac

m

products

m

o

r vSvSS



Efficiency

T

T

q

q

hot

cold

hot

cold −=

Maximum performance factor for refrigerator =w

q

TT

T

coldhot

cold =−

Helmholtz energy A = U – TS

Gibbs energy

G = H – TS

Standard reaction Gibbs energy ∆ r GO = Δr H

o – TΔr So

dU = TdS – PdV dG = VdP – SdT

Gibbs-Helmholtz equation

Chemical potential

Fundamental equation

Extent of reaction P T

r

GG

,

energy,reactionGibbs

∂∂=∆ξ

v

T

T1

q

w

hot

cold

hot

cycle −==−

=ε

∑∑ −=tstanreac

o f

products

o f r GvGvG ∆∆∆

−∆+

∆=

∆∆−=

∂∆∂

12

1

1

)1

2

)2

2

P T

1

T

1)T(H

T

T(G

T

T(G

T

H

T

)T/G(

∫ ∫ ===∆f

i

f

i

P

P i

f

P

PP

PlnnRT

P

dPnRTdP

P

nRTGgases,idealFor

)(,, i jn P T i

i

j

n

G

≠

∂∂

= µ

∑+= i

iidn μVdP -SdT dG

)/ln(gases,idealFor mbar P RT o

m += µ µ

o

o

P

f RTlngases,realFor tcoefficienfugacityPf Fugacity, +== µ µ φ φ

i

oii nn

υ ξ ,−=

ln o

P

o

r K RT G Δ −=Q RT GGo

r r ln+∆=∆



van’t Hoff equation

−−=

121

2 11ln

T T R

H Δ

K

K o

r

Rate laws

First order :kt

ot e A A −= ][][ Second order: kt A A o

=−][

1

][

1 kt A B

A A

B Boo

o

o )][]([]/[][

]/[][ln −=

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

Half-life

First-order:k k

t 693.02ln2/1 == Second-order:o Ak

t ][

12/1 = Zero-order:

Arrhenius equation:

Enzyme kinetics:

Clapeyron equation: m

m

ΔV

ΔS

dT

dP =

i

f

m

m

T

T ln

V

ΔH P

∆∆ =

Clausius-Clapeyron equation:

Raoult’s law: Psolvent = xsolventPosolvent

Henry’s law: PB = bBK B

Colligative properties:

Boiling point elevation: ∆ T = K bb

Freezing point depression : ∆ T = K f b

van’t Hoff equation: ΠV soln = n solute RT

vi

∑= υ )( RT K K c p

k At o

2][2/1 =

RT a E

Aek −

=

−

∆−=

i f

vap

i

f

T T R

H

P

P 11ln

]/[1

][2

S K

E k v

M

o

+=

][

111

maxmax S V

K

vv

M

+=

Documents

Formula sheet_Final