A Brief Review of Thermodynamics

Internal Energy and the First Law The infinitesimal change in the internal energy  For a general process The First Law of Thermodynamics

The Constant Volume Heat Capacity Define the constant volume heat capacity, C V

Enthalpy We define the enthalpy of the system, H

The Constant Pressure Heat Capacity Define the constant pressure heat capacity, C 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

The Fundamental Equation of Thermodynamics Combine the first law of thermodynamics with the definition of entropy.

The Temperature dependence of the Entropy Under isochoric conditions, the entropy dependence on temperature is related to C V

Entropy changes Under Constant Volume Conditions For a system undergoing an isochoric temperature change  For a macroscopic system

The Temperature dependence of the Entropy Under isobaric conditions, the entropy dependence on temperature is related to C P

Entropy changes Under Constant Pressure Conditions For a system undergoing an isobaric temperature change  For a macroscopic system

The Second Law of Thermodynamics The second law of thermodynamics concerns itself with the entropy of the universe (  univ S).  univ S unchanged in a reversible process  univ S 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

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=-P ext dV+dw a where dw a includes all other types of work

The General Condition of Equilibrium and Spontaneity For a general system

Isothermal Processes For a systems where the temperature is constant and equal to T surr

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

Isothermal Volume Changes For an ideal gas undergoing an isothermal volume change

Isothermal Processes at Constant Pressure For an isothermal, isobaric transformation

The Gibbs Energy Define the Gibbs energy G G(T,P) =U – TS+PV Note that for an isothermal process dG  dw a  G  w a For an isothermal, isobaric process  G  0

The Properties of G The Gibbs energy is a function of temperature and pressure

Isothermal Pressure Changes For an ideal gas undergoing an isothermal pressure change

Temperature Dependence of A Under isochoric conditions

Gibbs Energy Changes As a Function of Temperature The Gibbs energy changes can be calculated at various temperatures

The Chemical Potential Define the chemical potential  = G/n

Gibbs Energy and Spontaneity  sys G < 0 - spontaneous process  sys G > 0 - non-spontaneous process (note that this process would be spontaneous in the reverse direction)  sys G = 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 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  f H  = 0 kJ/mole at all temperatures S  = 0 J/(K mole) at all temperatures  f G  = 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 = n s  s + n  = n s  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 , a - and 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 - =  - m - a + =  + m + As with the activities (   ) = (  + ) + (  - ) - (m  ) = (m + ) + (m - ) -

The Chemical Potential Expression This can be factored into two parts The ideal part Deviations from ideal behaviour

KCl CaCl 2 H 2 SO 4 HCl LaCl 3 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, 63 rd ed.; R.C. Weast Ed.; CRC Press, Boca Raton, Fl (1982).

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.

u This is valid in the up to a concentration of molal! The Debye Hűckel Limiting Law Z + = charge of cation; z - = charge of anion

Debye Hűckel Extended Law This equation can reliably estimate the activity coefficients up to a concentration of 0.10 mole/kg. B = 1.00 (kg/mole) 1/2

The Davies Equation This equation can reliably estimate the activity coefficients up to a concentration of 1.00 mole/kg. k = 0.30 (kg/mole)