Download Chemistry 434 - St. Francis Xavier University

A Brief Review of Thermodynamics
Internal Energy and the First Law
 The infinitesimal change in the internal energy
dU  dq  dw
 For a general process
U   dq  dw  q  w
c
The First Law of Thermodynamics
The Constant Volume Heat
Capacity
 Define the constant volume heat capacity, CV
 dq   U 
CV      
 dT V  T V
Enthalpy
 We define the enthalpy of the system, H
H  U  PV
The Constant Pressure Heat
Capacity
 Define the constant pressure heat capacity, CP
 dq 
 H 
CP  



 dT P  T P
Thermodynamic Definition
 Spontaneous Process – the process occurs
without outside work being done on the system.
Mathematical Definition of Entropy
 The entropy of the system is defined as follows
dq rev
dS 
T
The Fundamental Equation of
Thermodynamics
 Combine the first law of thermodynamics with the
definition of entropy.
dq rev
1
dS 
 dU  PdV
T
T

The Temperature dependence of
the Entropy
 Under isochoric conditions, the entropy
dependence on temperature is related to CV
CV
 S 
 T   T

V
Entropy changes Under Constant
Volume Conditions
 For a system undergoing an isochoric temperature
change
CV
dS 
dT
T
For a macroscopic system
T2
CV
S  
dT
T T
1
The Temperature dependence of
the Entropy
 Under isobaric conditions, the entropy
dependence on temperature is related to CP
CP
 S 
 T   T

P
Entropy changes Under Constant
Pressure Conditions
 For a system undergoing an isobaric temperature
change
CP
dS 
dT
T
For a macroscopic system
T2
CP
S  
dT
T T
1
The Second Law of
Thermodynamics
 The second law of thermodynamics concerns
itself with the entropy of the universe (univS).
 univS unchanged in a reversible process
 univS always increases for an irreversible process
The Third Law of Thermodynamics
 The Third Law - the entropy of any perfect crystal
is 0 J /(K mole) at 0 K (absolute 0!)
 Due to the Third Law, we are able to calculate
absolute entropy values.
Combining the First and Second
Laws
 From the first law
dU  dq  dw
T surr dS  dU  dw
Pressure Volume and Other Types
of Work
 Many types of work can be done on or by chemical
systems.
 Electrical work.
 Surface expansion.
 Stress-strain work.
dw=-Pext dV+dwa
where dwa includes all other types of work
The General Condition of
Equilibrium and Spontaneity
 For a general system
dU  Pext dV  dw a  T surr dS  0
Isothermal Processes
 For a systems where the temperature is constant
and equal to Tsurr
dU  TdS  dw  0
The Helmholtz Energy
 Define the Helmholtz energy A
A(T,V) =U – TS
 Note that for an isothermal process
dA  dw
A  w
 For an isochoric, isothermal process
A  0
The Properties of A
 The Helmholtz energy is a function of the
temperature and volume
 A 
 A 
dA  
dT  
dV


 T V
 V T
 A 
 T   S

V
 A 
 V   P

T
Isothermal Volume Changes
 For an ideal gas undergoing an isothermal volume
change
V2
V2
 A 
A   
dV   PdV

V1  V T
V1
V 2 
 nRT ln  
V1 
Isothermal Processes at Constant
Pressure
 For an isothermal, isobaric transformation
dU  TdS  Pext dV  dw a  0
dU  TdS  PdV  dw a  0
The Gibbs Energy
 Define the Gibbs energy G
G(T,P) =U – TS+PV
 Note that for an isothermal process
dG  dwa
G  wa
 For an isothermal, isobaric process
G  0
The Properties of G
 The Gibbs energy is a function of temperature
and pressure
 G 
 G 
dG  
dT  
dP


 T P
 P T
 G 
 T   S

P
 G 
 P   V

T
Isothermal Pressure Changes
 For an ideal gas undergoing an isothermal
pressure change
P2
V2
 G 
G   
dP  VdP

P1  P T
V1
 P2 
 nRT ln  
 P1 
Temperature Dependence of A
 Under isochoric conditions
 A 
dA  
dT  SdT

 T V
T2
T2
 A 
A   
dT   SdT

T1  T V
T1
Gibbs Energy Changes As a
Function of Temperature
 The Gibbs energy changes can be calculated at
various temperatures
G T 2   G T1  
T2
T S
dT
1
G T2   G T1   S T 2 T1 
The Chemical Potential
 Define the chemical potential  = G/n
 P2 
G 2  G1

 RT ln  
n
n
 P1 
G
 P2 
2  1  RT ln  
 P1 
Gibbs Energy and Spontaneity
sysG < 0 - spontaneous process
sysG > 0 - non-spontaneous process (note that this
process would be spontaneous in the reverse
direction)
sysG = 0 - system is in equilibrium
Applications of the Gibbs Energy
 The Gibbs energy is used to determine the
spontaneous direction of a process.
 Two contributions to the Gibbs energy change (G)
 Entropy (S)
 Enthalpy (H)
G = H - TS
Thermodynamics of Ions in
Solutions
    RT ln m
 Electrolyte solutions – deviations from ideal
behaviour occur at molalities as low as 0.01
mole/kg.
 How do we obtain thermodynamic properties of
ionic species in solution?
For the H+(aq) ion, we define




fH = 0 kJ/mole at all temperatures
S = 0 J/(K mole) at all temperatures
fG = 0 kJ/mole at all temperatures
Activities in Electrolyte Solutions
 For the following discussion
 Solvent “s”
 Cation “+”
 Anion “=“
 Consider 1 mole of an electrolyte dissociating
into + cations and - anions
G = ns s + n 
= ns s + n+ + + n-  Note – since  = + + -   = + + + - -
The Mean Ionic Chemical Potential
 We define
 =  / 
 We now proceed to define the activities
 =  + RT ln a
+ = + + RT ln a+
- = - + RT ln a =  + RT ln a
The Relationship Between a and a
 Since   =  / 
 =  + RT ln a =  ( + RT ln a)
Since  =  / 
 This gives us the relationship between the electrolyte
activity and the mean activity
(a)= a
The Relationship Between a , aand a+
 We note that  = + + + - -
and   =  / 
 This gives us the following relationship
( + RT ln a) = + (+ + RT ln a+) +
- ( - + RT ln a-)
 Since   = + + + - -
(a) = (a+)+ (a-)-
Activities in Electrolyte Solutions
 The activities of various components in an
electrolyte solution are defined as follows
a+ = + m+
a- = - ma+ = + m+
 As with the activities
() = (+)+ (-)(m) = (m+)+ (m-)-
The Chemical Potential Expression
    RT ln a     RT ln  m    
    RT ln  m        RT ln  m     RT ln   


    RT ln  m   RT ln        RT ln    
 This can be factored into two parts

     RT ln m   RT ln  
The ideal part
 


   RT ln  

Deviations from
ideal behaviour
Activity Coefficients As a Function of Molality
Data obtained from
Glasstone et al., Introduction to Electrochemistry, Van Nostrand (1942).
CRC Handbook of Chemistry and Physics, 63rd ed.; R.C. Weast Ed.; CRC Press,
Boca Raton, Fl (1982).
CaCl2
HCl
LaCl3
KCl
H2SO4
Estimates of Activity Coefficients in
Electrolyte Solutions
 The are a number of theories that have been
proposed to allow the theoretical estimation of the
mean activity coefficients of an electrolyte.
 Each has a limited range of applicability.
The Debye Hűckel Limiting Law
 This
is valid in the up to a
concentration of 0.010 molal!
log    0.510 z  z   Im 
1
2
Z+ = charge of cation; z- = charge of anion
I m  Ionic strength  ½
j  z j  m j
2
Debye Hűckel Extended Law
 This equation can reliably estimate the activity
coefficients up to a concentration of 0.10
mole/kg.
log   
0.510 z  z   I c 
1  B I c

B = 1.00 (kg/mole)1/2
1
2
1
2
The Davies Equation
 This equation can reliably estimate the activity
coefficients up to a concentration of 1.00
mole/kg.
log    0.510 z  z 
1


2
 I m 


kI
m 
1

1   I m  2

k = 0.30 (kg/mole)