1) Gibbs phase rule determines the number of intensive properties (F) that can be independently varied for a system with N chemical species and P phases. For a single phase of a pure substance, F=2. For two coexisting phases, F=1. For three coexisting phases, F=0. 2) The lever rule determines the mole fraction of each phase in a binary equilibrium phase diagram. It relates the composition of an alloy to the compositions of its constituent phases. 3) The lever rule equation for the weight percentage of an α phase is: Xα = (c - b) / (a - b), where a, b, c are the weight percentages of an element in

1. GIBBS PHASE RULE AND
LEVER RULE
Material Science and Metrology

2. Gibbs Phase Rule
• Consider a glass of water. This is a pure substance (H20) in a single
phase. It can be described with a number of thermodynamic properties
including temperature, pressure, volume, entropy, enthalpy and Gibbs
energy. However not all of these properties are independent. In fact,
for the glass of water, only two intensive thermodynamic properties
are independent. Thus, if we specify temperature and pressure, all the
other properties, in their intensive (i.e. their value per mole of
substance) form, can be determined.

3. Temperature and pressure are often taken as independent intensive variables.(see P-
T diagram) This is because it is usually easy to experimentally vary and measure
these two properties directly. However, any intensive property can be choosen to be
one of the independent variables. If the intensive enthalpy (J/mol) and intensive
entropy (J/mol K) of the water in the glass were specified, its temperature and
pressure could then be found using the steam tables or a Mollier diagram.

4. • The Gibbs phase rule tells how many independent intensive properties, F, can be
chosen. This will depend on the number of chemical species, N, and number of
phases, pi, present. In the absence of chemical reaction, the Gibbs phase rule is
simply:
single phase (pi=1) F=2+1-1=2
two phases (pi=2) F=2+1-2=1
three phase (pi=3) F=2+1-3=0
F=2+N-pi
For a pure substance (N=1), the Gibbs phase rule can be applied as
follows:

5. An example showing that for a single phase of a pure
substance, F=2:
• For a glass of liquid water, specify one of the independent intensive
variables to be pressure. Choose this pressure to be 1 atm. If liquid is
in the glass, the temperature can take any value between 0 'C and 100
'C. Within this range, the temperature can be choosen independently of
the pressure. Thus, both T and P are independent. After choosing a (T,
P) pair, any other property, such as volume or entropy, can be found
using the steam tables or a Mollier diagram. Therefore the remaining
properties can not be independently choosen after T and P are
specified.

6. Example showing that for two coexistant phases of a pure
substance, F=1.
• For a glass of boiling water (also call saturated liquid water) in
equilibrium with saturated steam, specify the one of the independent
intensive variables to be pressure. Choose this pressure to be 1 atm. In
order for the water to boil (or be saturated) at this pressure, the
temperature must be 100 'C. Thus, the temperature can not be choosen
independently of the pressure when both liquid and vapor water are
present. Thus, only the P is independent. The temperature required at
other pressures, as well as the values of all remaining thermodynamic
properties, can be found in the tables for saturated steam or on a
Mollier diagram.

7. Example showing that for three coexistant phases of a pure
substance, F=0.
• At the triple point; vapor, liquid and solid all coexist. For any given
substance, the triple point occurs only at one a specific pair of
temperature and pressure. Once it is stated the substance at the triple
point, the values of this temperature and pressure pair as well as the
values of all other thermodynamic properties can be found in a table or
graph. Thus, no thermodynamic property can be choosen
independently.

8. Lever rule
The lever rule is a tool used to determine mole fraction of each phase of a binary
equilibrium phase diagram. It is used to determine the percent of liquid and solid
phases for a given binary composition and temperature that is between the liquidus
and solidus line.
In an alloy with two phases, α and β, which themselves contain two elements, A and
B, the lever rule states that the weight percentage of the α phase is
𝑋∝ =
(𝑐 − 𝑏)
(𝑎 − 𝑏)
where
• a is the weight percentage of element B in the α phase
• b is the weight percentage of element B in the β phase
• c is the weight percentage of element B in the entire alloy
all at some fixed temperature.

9. Binary phase diagrams
Before any calculations can be made, a tie line is drawn on the phase diagram to
determine the percent weight of each element; on the phase diagram to the right it is
line segment LS. This tie line is drawn horizontally at the composition's temperature
from one phase to another (here the liquid to the solids). The percent weight of
element B at the liquidus is given by wl and the percent weight of element B at the
solidus is given by ws. The percent weight of solid and liquid can then be calculated
using the following lever rule equations:
• percent weight of solid phase : 𝑋𝑠 =
wo− wl
ws− wl
• percent weight of liquid phase : 𝑋𝑙 =
ws−wo
ws− wl
where wo is the percent weight of element B for the given composition.

10. The numerator of each equation is the original composition we are interested in +/-
the Opposite lever arm. That is if you want the percent weight of solid then take the
difference between the liquid composition and the original composition. And then
the denominator is the overall length of the arm so the difference between the solid
and liquid compositions.

11. THANK YOU