Thermodynamics: the background in a Nutshell
Notation: changes, sums and integrals
Infinitesimal changes (i.e. so small they are almost zero) denoted by “d”
e.g. temperature: dT; internal energy: dU; pressure: dP …
Actual changes (i.e. big enough to measure) denoted by “”
e.g. temperature: T; internal energy: U; pressure: P …
An actual change can be thought of as made up of an infinite number of infinitesimal changes. The
actual change is then the sum of all these infinitesimal changes. We call this kind of sum “an
integral”
T final
Tfinal – Tinitial = T = T final  Tinitial  T 

dT
Tinitial
Rules of Integration:
 dx  x
1
 x dx   x
1
1
 x dx  n x
n
e
ax
dx 
dx  ln x
n 1
(any value of n except -1)
1 ax
e
a
"definite integrals": if
 f ( x)dx  F ( x)
then

b
a
f ( x)dx  F (b)  F (a )
Notation: intensive and extensive variables
Thermodynamic properties can be divided into two classes: those (like temperature or pressure) that
are independent of how much material is present, which are called intensive variables; and those
(like volume or energy) which are proportional to the amount of material present — and are called
extensive variables.
Extensive variables are often converted to intensive by dividing by the molar amount (to give the
“molar energy” or “molar volume”).
IUPAC recommendations: internal energy (volume, Gibbs free energy, etc.) are denoted by E (V, G)
Molar quantities have a subscript “m” added (hence Em, Vm, Gm …)
SI units for E or G are be J (kJ etc.), while those for Em are J mol–1
The book used for this course adopts a different convention, where the original quantity carries a
superscript t for “total”, and the molar quantity is unmodified. Hence
Internal energy E t, but molar internal energy would be E.
(This may seem confusing, but if you include your units explicitly at every step of your calculation,
they will take care of the problem for you!)
Thermodynamics in a Nutshell
More in depth material available at: http://go.warwick.ac.uk/molsaw/pmrodger/pmrteach/pcbio/, or follow the link to
“teaching material”, for CH932 from my page (P.M. Rodger) on the Chemistry staff web pages.
State of a system

Defined by quantities such as temperature (T), pressure (P), entropy (S) and volume (V)

Only two of these can be specified: the other two then follow from the properties of the
system (“equation of state”)
e.g. ideal gas:

PV = nRT
Usually choose one of T or S, and one of P or V
Types of Energy
name
Symbol
Definition
Changes
internal energy
U (or E)
enthalpy
H
H = U + PV
dH = TdS + VdP
Gibbs free energy
G
G = H – TS
dG = –SdT + VdP
dU = TdS – PdV
Heat & Work
“heat” (q or Q) measures energy changes that are driven by changes in temperature
Heat capacity determines how the energy contained in a system/substance changes with
temperature. Its value depends on whether the temperature is changed at fixed volume
(isochoric, CV) or at fixed pressure (isobaric, CP)
 U 
CV  

 T V

qV  nCV T
 H 
CP  

 T V

qP  nCP T
“work” (w) measures energy changes that are driven by changes in volume. It depends on
how hard the system has to push to make that volume change.
w    Pexternal dV
1st law of thermodynamics
Energy is conserved. Thus, changes in internal energy must be from the heat and work
U = q + w
2nd law of thermodynamics
Heat is related to entropy. If heat is applied reversibly (i.e. making very slow infinitesimal changes
and never disturbing equilibrium) then entropy defines heat changes
S

qreversible
q
T
 TdS