Physical Chemistry
Lecture 25
Hess’s Rule, Temperature Dependence
of Enthalpy of Reaction, and Adiabatic
Flame Temperature Calculations
Thermodynamics of reaction
processes
Often faced with the need to calculate the
change of a thermodynamic variable in a
reaction that may not be tabulated
C2 H 4 ( g ) + H 2 ( g ) → C2 H 6 ( g )
N 2O ( g ) +
3
O2 ( g ) → N 2O4 ( g )
2
Applies to all thermodynamic variable
∆H reaction
∆S reaction
∆U reaction
∆Areaction
∆Greaction
Cyclic properties of
thermodynamic functions
Consider a cycle involving
reaction
Change in a thermodynamic
variable


Change through a cycle is
zero
Change along one path
between two states must
equal that along another
connecting the two states
True for ∆H, ∆U, ∆S, ∆A, ∆G
Hess’s rule
The cyclic property involving reaction is
codified in Hess’s rule
∆H reaction
=
∑ν
k
∆ f H k ,m
k
True of any state function
∆U reaction
=
∑ν
k
∆ f U k ,m
k
∆S reaction
=
∑ν
k
∆ f S k ,m
∑ν
k
∆ f Ak ,m
k
∆Areaction
=
k
∆Greaction
=
∑ν
k
k
∆ f Gk ,m
Hess’s rule and combustions
The cyclic property is also true when the
intermediate state is the stable oxides
∆H reaction
= − ∑ν k ∆H k ,m , combustion
k
∆U reaction
= − ∑ν k ∆U k ,m , combustion
k
∆S reaction
= − ∑ν k ∆S k ,m , combustion
k
∆Areaction
= − ∑ν k ∆Ak ,m , combustion
k
∆Greaction
= − ∑ν k ∆Gk ,m , combustion
k
Example: Boudouard’s reaction
Reaction
C ( gr )
+
CO2 ( g )
→
2 CO ( g )
By Hess’s rule
θ
∆H reaction
(298.15 K ) = 2 ∆ f H mθ (CO ) − ∆ f H mθ (C ) − ∆ f H mθ (CO2 )
From Table 6.1, one can find the enthalpies of
formation
θ
∆H reaction
(298.15 K ) = 2 (−110.525 kJ ) − (0 kJ ) − (−393.509 kJ )
= 172.459 kJ
Enthalpies and energies of formation of
elements are zero, by definition
Temperature dependence of
thermodynamic quantities
Chemical reactions are often carried out at
temperatures other than those for which information
is tabulated
Use cyclic properties to determine changes in
thermodynamic properties at other temperatures
Temperature dependence of enthalpy
of reaction
Enthalpy change is independent of the path
∆H reaction (T2 ) = ∆H1
+ ∆H reaction (T1 ) + ∆H 2
The two steps are simply heating of materials
∆H reaction (T2 ) =
T1
∫C
P , reactants
(T )dT
+ ∆H reaction (T1 ) +
T2
= ∆H reaction (T1 ) +
T2
∫C
P , products
T1
T2
∫ ∆C
P
(T )dT
T1
Evaluation of the heat-capacity difference allows estimate of
enthalpy change at different temperature
(T )dT
Temperature dependence of
thermodynamic quantities
Determining temperature dependence
involves evaluation of change for both
reactants and products
T2
θ
C
∆
∫ P (T )dT
θ
θ
(T2 ) = ∆H reaction
(T1 ) +
∆H reaction
T1
T2
θ
C
∆
∫ V (T )dT
θ
θ
(T2 ) = ∆U reaction
(T1 ) +
∆U reaction
T1
θ
C
∆
θ
θ
P (T )
dT
(T2 ) = ∆S reaction
(T1 ) + ∫
∆S reaction
T
T1
T2
Adiabatic flame temperature
Question: What is the
temperature in a flame?
Can answer this
question in a limit


Heat generated raises
temperature of the
products
No losses of heat –
adiabatic process
Solve for T2, adiabatic
flame temperature


Can be difficult
because it may be a
complex equation in T2
Must know what
materials are
“products”
T2
θ
C
∫ P, products (T )dT
T1
θ
= − ∆H combustion
(T1 )
Adiabatic flame temperatures
Fuel
Oxidizer
Reaction
Tad (K)
Hydrogen
Oxygen
H2 + ½ O2  H2O
3100
Methane
Oxygen
CH4 + 2 O2  CO2 + 2 H2O
3000
Methane
Air
CH4 + 2 O2 + 8 N2  CO2 + 2 H2O + 8 N2
2200
Octane
Oxygen
C8H18 + 25/2 O2  8 CO2 + 9 H2O
3100
Acetylene
Oxygen
C2H2 + 5 O2  4 CO2 + 2 H2O
3300
Cyanogen
Oxygen
C2N2 + 5 O2  2 CO2 + N2
4800
Producer gas
Air
2 CO + 4 H2 + 3 O2 + 12 N2  2 CO2 + 4 H2O + 12 N2
2400
Methylhydrazine
Nitrogen
tetroxide
CH2NH2 + N2O4  2 H2O + CO2 + 2 N2
3000
Summary
Hess’s rule allows estimation of
thermodynamic changes for a wide variety
of chemical reactions
Use of thermodynamic cycles allows
estimation of thermodynamic changes at
conditions other than those in tables,
particularly at other temperatures
Adiabatic flame temperature calculation
relies on a similar cyclic calculation