Peter Atkins • Julio de Paula
Atkins’ Physical Chemistry
Eighth Edition
Chapter 2 – Lecture 4
The First Law
Copyright © 2006 by Peter Atkins and Julio de Paula
State Functions and Exact Differentials
• State function – state of the system independent of manner
and time to prepare
• Path functions – processes that describe prep of the state
• e.g., Heat, work, temperature
• Systems do not possess heat, work, or temperature
• Advantage of state functions:
• Can combine measurements of different properties to
obtain value of desired property
Fig 2.20 As V and T change, the internal energy, U changes
non-adiabatic
adiabatic
When the system
completes the path:
f
ΔU 
 dU
i
dU is said to be:
an exact differential
f
q
 dq
i,path
q is said to be:
an inexact differential
Fig 2.21 The partial derivative (∂U/∂V)T is the slope of U
w.r.t V at constant T
Fig 2.22 The partial derivative (∂U/∂T)V is the slope of U
w.r.t T at constant V
Fig 2.23 The overall change in U (i.e. dU) which arises when
both V and T change
 U 
 U 
dU  
 dV  
 dT
 V  T
 T  V
The full differential of U
w.r.t V and T
Partial derivatives representing physical properties:
 U 
Constant volume heat capacity C V  

 T  V
 H 
Constant pressure heat capacity CP  

 T  P
Internal pressure
 U 
πT  

 V  T
Fig 2.24 The internal pressure πT is the slope of U w.r.t. V
at constant T
has dimensions of pressure
Partial derivatives representing physical properties:
 U 
Constant volume heat capacity C V  

 T  V
 H 
Constant pressure heat capacity CP  

 T  P
Internal pressure
Now:
 U 
πT  

 V  T
 U 
 U 
dU  
 dV  
 dT  π T dV  C V dT
 V  T
 T  V
Fig 2.25 For a perfect gas the internal energy is independent
of volume at constant temperature
Two cases for real gases:
 U 
πT  
 0
 V  T
Fig 2.26 Schematic of Joule’s attempt to measure ΔU for
an isothermal expansion of a gas
q absorbed by gas ∝ T
 U 
πT  

 V  T
• For expansion into vacuum,
w=0
• No ΔT was observed,
so q = 0
at 22 atm
• In fact, experiment was crude
and did not detect the
small ΔT
Changes in Internal Energy at Constant Pressure
• Partial derivatives useful to obtain a property that cannot
be observed directly:
• e.g.,
dU  π T dV  CV dT
• Dividing through by dT:
 U 
 V 

  πT 
  CV
 T  P
 T  P
1  V 
• Where α  
 expansion coefficient
V  T  P
1  V 
 isothermal compressibility
• and κ T   
V  P  T
Changes in Internal Energy at Constant Pressure
1  V 
Substituting: α  

V  T  P
Into:  U   π T  V   C V
 T  P
 T  P
Gives:  U   απ V  C
T
V
 T  P
For a perfect gas:
 U 

  CV
 T  P
1  V 
κT   

V  P  T
Changes in Internal Energy at Constant Pressure
• May also express the difference in heat capacities with
observables:
• Cp – Cv = nR
 H 
 U 
CP  C V  
 

 T  P  T  P
• Since H = U + PV = U + nRT
 U 
 H 
CP  C V  
  nR  
  nR
 T  P
 T  P
• And:
α 2 TV
CP  C V 
κT