Exercise no. 1

BINARY SOLID-LIQUID PHASE DIAGRAM

Jomari R. Noe; Ranelle dP. Acda; Rommel Yecla; Marielle A. Leysa; Maris L. Bayhon

Date of performance: 10 August 2015; Date submitted: 24 August 2015; Laboratory section:

CHEM 112.1 2L; Instructor: Anabelle T. Abrera

Abstract

A binary solid-liquid phase diagram has been constructed

from the cooling curves of mixtures containing varying amounts of

diphenylamine and naphthalene. The cooling curves were obtained

from heating the mixtures until molten and then cooling slowly.

From each of the cooling curve, the break and arrest temperatures

were determined. These temperatures were plotted against the mole fraction of naphthalene in each mixture. From the constructed

phase diagram, the eutectic composition was graphically

determined. ΧA is equal to 0.5190 while ΧB is equal to 0.4810. The

eutectic temperature, calculated from the average of the arrest

temperatures is equal to 25.83°C. The theoretical values of ΧA and

ΧB calculated using the Newton-Raphson method were 0.6419 and

0.3581, respectively. The theoretical eutectic temperature, on the

other hand, was 32.44°C.

I. Introduction

A phase is a form of matter that is

homogeneous, uniform in chemical composition, and physically distinct all

throughout. In a heterogeneous system,

phases can be defined by boundaries.

(Atkins, et al., 2006; Sivasankar, 2008). Solid,

liquid, and gas phases are the most common

phases of matter.

Matter can be converted from one

phase to another. This process is called phase transition. Examples of phase

transition include gas to liquid

(condensation) or liquid to solid (freezing).

Phase transitions can suddenly and

spontaneously take place, in certain

temperature at a given pressure. For

instance, if a gas is a little above its

condensation point, a slight decrease in

temperature yields a liquid phase that

coexists with the gas phase. If the

temperature is decreased further, the system

which is initially gas becomes a single liquid

phase (Mortimer, 2008).

At a particular condition, maintaining

certain temperature and pressure, two or

more phases can coexist, as mentioned

previously (Monk, 2004). Normally, water

boils at 100°C. At this temperature, vapour

pressure is equal to the atmospheric

pressure. Moreover, the liquid and gas states

of water coexist at equilibrium. This existing

equilibrium is dynamic – that is, as liquid is

transforms into gas, an equal quantity is also

converted back to the liquid state.

Pressure and temperature values are

dependent on each other, therefore, a

change in pressure, equilibrium temperature

shifts consequently. By plotting the

experimental values of pressure and

temperature at which equilibrium occurs, we

obtain a phase diagram.

Figure 1.1 Phase diagram of water (Source:

Atkins & de Paula, 2006)

This diagram is only for a one-

component system. A component can be

defined as an independent chemical

component. “The number of components in

a system refers to the smallest number of

independently variable chemical

constituents by which the composition of

each phase can be expressed directly or

through an equation (Sivasankar, 2008).”

The Gibbs phase rule gives the

number of independent intensive variable in

a system that may have several phases and

components (Mortimer, 2008). It also

provides the relationship between the

degrees of freedom of a system f, the

number of phases p, and the number of

components c (Moore, 1962).

Mathematically, the Gibbs phase rule is stated as:

f = c – p + 2 (eq. 1.1)

The constant 2 suggests that the

temperature and pressure are specified.

In a system consisting of ice, water,

and water vapour, the number of phases is

three (i.e., solid, liquid, and gas, respectively).

The number of component, on the other

hand, is one since the system can be

represented using a single chemical

constituent: water. Moreover, using Gibbs

phase rule, the number of degrees of

freedom is equal to:

f = 1 - 3 + 2 = 0

The system above in invariant (f = 0)

since the three phases of water coexist.

Moreover, neither the temperature nor the

pressure can be varied slightly without

causing one of the phase to disappear.

Looking at the phase diagram of water, the

point where the system in invariant is located at the triple point.

When 2 phases are only present, the

system is univariant. Along the boundary

lines of the phase diagram of water, the

pressure can be varied with temperature

such that two phases may coexist.

Finally, when 1 phase is only present,

the system is bivariant. Pressure and

temperature can be varied independently

while maintaining only a single phase.

In this experiment, a mixture of

naphthalene and diphenylamine is used. This number of components in this system is

two (c = 2) since each component behaves

independently and one component cannot

be represented by the other. Moreover, the

concentrations of naphthalene and

diphenylamine can be varied independently

in various phases (Moore, 1962).

Such system may also be represented through phase diagrams,

particularly, a binary solid-liquid phase

diagram.

In constructing the diagram for this

system, the pressure may be held constant,

thus, reducing equation 1.1 into:

f = c – p + 1 (eq. 1.2)

f = 3 - p

The method of thermal analysis is

usually done to construct binary phase

diagrams for eutectic systems like the

naphthalene-diphenylamine system.

In thermal analysis, a mixture with

known composition is heated until all solids

have melted. The mixture is allowed to slowly

cool and a cooling curve is constructed

(Mortimer, 2008). From these cooling curves,

break temperatures are obtained and plotted against the composition of the mixture. On

the other hand, the mean arrest

temperatures of the cooling curves is

calculated and designated as the eutectic

temperature. In the phase binary phase

diagram, a straight horizontal line is drawn at

this temperature. Shown below is an

example of binary phase diagram of eutectic

systems.

Figure 1.2 Binary phase diagram for

naphthalene and benzene (Source: Moore,

1962)

The figure is a binary diagram for

naphthalene-benzene system. The CE curve

marks the boundary where the first solid

naphthalene appears. On the other hand, the

curve DE marks the point where benzene

starts to solidify. These curves are referred

to as liquidus curves. Above these curves,

the system is entirely liquid.

The dashed line in the phase diagram

represents the solidus curve, below which

solid naphthalene and solid benzene coexist

and the liquid or melt naphthalene and

benzene disappear.

Point E is referred to as the eutectic

point. It is where the two liquidus curve meet.

This point is characterized by the eutectic

temperature and the eutectic temperature

(Patra and Samantray, 2011).

This exercise aims to construct a

solid-liquid phase diagram for a simple, non-

reacting binary system and interpret this diagram in terms of the eutectic composition,

the number of components, the number and

nature of phases present and the number of

degrees of freedom.

II. Methodology

In a large test tube, specific amounts

of naphthalene and diphenylamine were

added. The amounts were based on Table

1.1 of the laboratory manual.

Eight runs were commenced in the

experiment. For each run, the large test tube

was heated in a water bath until all the solid inside the tube has melted or a certain

temperature is reached. For runs 1-5, 90°C

was used while 62 °C was used for runs 6-8.

After all the solids have melted, the

test tube was inserted inside the Dewar flask

fill with crushed ice.

The temperature was recorded for

every 10 seconds until 20°C is reached. The

process is repeated for every run.

Only one trial was done due to time

constraints.

Mixtures are disposed in waste

bottles labelled “Non-halogenated”. On the

other hand, the test tube and stirrer were

washed with acetone and the washing was

disposed in the same container.

III. Results and Discussion

Eight heating runs were made in the

experiment. For each run, a cooling curve

has been generated. Also, each run contains

varying amounts of (B) naphthalene and (A)

diphenylamine.

Table 1.1 Amount of A and B in each run

Run no. Amount, g

1 5.0292 B 2 1.0002 A 3 1.5018 A 4 2.5003 A 5 5.0020 A 6 5.0003 A

7 1.0007 B 8 0.6707 B

A cooling curve can be constructed

by reversibly cooling a melted or liquid

substance (Kaurav, 2011). It shows how the

temperature of a substance falls with time. It

also exhibits the point at which temperature

does not change through some period of time (Kakani, 2004). The constant

temperature is referred to as the arrest

temperature.

The cooling curve of a pure

substance slightly differs with that of a

mixture. In both cases, an arrest temperature

can be observed. However, the rate of

freezing is different.

For a pure substance, like

naphthalene, the rate of freezing remains

constant before it reaches the arrest

temperature. On the other hand, the rate of

freezing in a melted naphthalene-

diphenylamine mixture changes at some

point before reaching the arrest temperature.

This is caused by the change in the

concentration of the liquid mixture. As the

mixture cools down, one component begins

to solidify while the other remains in the liquid

state.

Shown below are the cooling curves

for a silver-copper system. The differences in

the cooling curves are associated on the

composition of each of the components in

the mixture.

Figure 1.3 Cooling curves for the silver-

copper system (Source: Mortimer, 2008)

In this experiment, a binary solid-

liquid phase diagram was constructed from

the cooling curve of each of the run. The

resulting phase diagram is shown below.

Two points were omitted from the set of data

to generate a better phase diagram.

Figure 1.4 Constructed phase diagram for

the naphthalene-diphenylamine system

The different regions in the phase

diagram are labelled I, II, III, and IV. I is

composed of the molten naphthalene and

diphenylamine. This region is only composed

of one phase, the homogenous molten

mixture, since diphenylamine and naphthalene are miscible in the liquid phase.

Therefore, using equation 1.1, f is equal to 2.

Region II is composed of solid

naphthalene and the molten mixture. The

number of phases in this region is 2.

Therefore, f in this region is 1. Region III has

the same value of f – it is composed of solid

diphenylamine and the molten mixture.

0

10

20

30

40

50

60

70

80

0 0.2 0.4 0.6 0.8 1

I

II

III

IV

e

Region IV consists of solid

diphenylamine and solid naphthalene. The

solid phases are not miscible with each other

so p is equal to 2. Therefore, the value of f is

1.

The point labelled with e is the

eutectic point. It is at this point that the

molten mixture and the two solid phases

coexist. At this point, the number of phases

is equal to 3. So, f is 0.

The arrest temperature in each run

was obtained and their average was

calculated. The calculated temperature was

25.83 °C. This temperature corresponds to

the eutectic temperature.

Also, at this temperature, the eutectic

composition was determined graphically –

that is, 0.481 diphenylamine and 0.519 naphthalene.

The theoretical composition and

temperature can also be determined through

calculations. The following equations were

used for the calculation.

ln 𝜒𝐴 = Δ𝑓𝑢𝑠 𝐴𝐻

𝑅(

1

𝑇𝑚,𝐴−

1

𝑇𝑒) (𝑒𝑞. 1.3)

ln 𝜒𝐵 = Δ𝑓𝑢𝑠 𝐵𝐻

𝑅(

1

𝑇𝑚,𝐵−

1

𝑇𝑒) (𝑒𝑞. 1.4)

where:

A – diphenylamine

B -- naphthalene

From these equations, a single

function was derived.ln

ln (1 − 𝜒𝑒)𝐶

𝜒𝑒𝐷

+ 𝑌 = 0 (𝑒𝑞. 1.5)

where:

C = Δ𝑓𝑢𝑠 𝐴𝐻

𝑅

D = Δ𝑓𝑢𝑠 𝐵𝐻

𝑅

Y = 1

𝑇𝑚,𝐵−

1

𝑇𝑚,𝐴

The theoretical eutectic composition

was first calculated through Newton-

Raphson method.

𝑥𝑛+1 = 𝑥𝑛 −𝑓(𝑥)

𝑓′(𝑥)

The calculated mole fraction form the

Newton-Raphson method was substituted to equation 1.2 and the theoretical eutectic

temperature was calculated. The following

table lists the experimental and calculated

values for the eutectic temperature and

composition, as well as the percent error.

Table 1.2 Summary table for the eutectic

temperature and composition

Parameter Theoretical Experimental % Error

𝜒𝐴 0.6419 0.5190 19.15%

𝜒𝐵 0.3581 0.4810 34.32%

Temperature 32.44 °C 25.83 °C 20.37%

The eutectic composition has the

lowest melting temperature than any

possible composition of the components in a

binary system. Its melting temperature is lower than that of the individual components.

This temperature is referred to as the

eutectic temperature (Patra and Samantray,

2011).

The eutectic composition is very

significant in fields such as metallurgy and

material science. In foundry and casting, it is

the vital point since it has the lowest temperature possible for the system. It also

helps in reducing energy cost.

Other applications of the eutectic

composition includes freezing mixtures,

eutectic alloys for soldering purposes (e.g.,

Pb and Sn). In inkjet printers, eutectic

mixtures are also used as inks.

Aside from cooling curve data, phase

diagrams may also be constructed through x-

ray diffraction methods. It has become a

standard method in phase identification in

equilibrium studies (Hummel, 1984). This

method is used to establish loci of phase

boundaries and to determine gram atomic

volumes (Zhao, 2007).

Some studies that made use of x-ray

diffraction in establishing phase diagrams

include the investigation of a quasi-binary

system LiInSe2-CuInSe2 (Weise, et al., 1996)

and investigation of the phase equilibria in

CdI2-Bi2O3 (Vassilev, et al., 2004) and

GeSe2-SnTe systems (Vassilev, et al., 2003)

IV. Conclusions

A phase diagram has been

successfully constructed for the naphthalene-diphenylamine system. The

constructed phase diagram also has a

distinguishable eutectic point.

The experimental eutectic

composition and temperature values are

relatively near the calculated theoretical

values. Errors may have risen from the

calculation of the mole fraction for the graphical construction of the diagram. During

the experiment, some naphthalene and/or

diphenylamine adhere to the sides of the test

tube, affecting the composition of the mixture

during heating and cooling.

V. References

Atkins, P & J de Paula. 2006. Physical Chemistry, 8th ed. NY: Oxford

University Press.

Hummel, FA. 1984. Introduction to Phase

Equilibria in Ceramic Systems. CRC

Press.

Kakani, SL & A Kakani. 2006. Material

Science. New Delhi: New Age

Internationa (P) Ltd., Publishers.

Kaurav, MS. 2011. Engineering Chemistry

with Laboratory Experiments. New

Delhi: PHI Learning Private Limited.

Monk, P. 2004. Physical Chemistry:

Understanding Our Chemical World.

England: John Wiley & Sons Ltd.

Moore, WJ. 1962. Physical Chemistry, 4th ed.

USA: Longmans Green and Co. Ltd.

Mortimer, RG. 2008. Physical Chemistry, 3rd

ed. USA: Elsevier Academic Press

Patra, BB & B Samantray. 2011. Engineering

Chemistry I. India: Dorling Kindersley Pvt. Ltd.

Sivansankar, B. 2008. Engineering

Chemistry. New Delhi: Tata McGraw-

Hill Publishing Company Limited.

Zhao, JC. 2011. Methods for Phase Diagram

Determination. UK: Elsevier BV.

VI. Calculations

Mole Fraction (A: Diphenylamine, B: Naphthalene)

Run 1:

𝜒𝐵 =𝑛𝐴

𝑛𝐴 + 𝑛𝐵=

5.0292𝑔128.16𝑔/𝑚𝑜𝑙

0𝑔169.22𝑔/𝑚𝑜𝑙

+ 5.0292𝑔

128.16𝑔/𝑚𝑜𝑙

1.00

𝜒𝐴 = 1.00 − 𝜒𝐵 = 1.00 − 1.00 = 0

Run 2:

𝜒𝐵 = 0.8690950608

𝜒𝐴 = 0.1309049392

Run 3:

𝜒𝐵 = 0.7263318323

𝜒𝐴 = 0.2736681677

Run 4:

𝜒𝐵 = 0.5703509791

𝜒𝐴 = 0.4296490209

Run 5:

𝜒𝐵 = 0.3989950942

𝜒𝐴 = 0.6010049058

Run 6:

𝜒𝐴 = 1.00

𝜒𝐵 = 0

Run 7:

𝜒𝐴 = 0.7909858284

𝜒𝐵 = 0.2090141716

Run 8:

𝜒𝐴 = 0.693793818

𝜒𝐵 = 0.306206182

Experimental Eutectic Temperature

𝑇𝑒 = ∑ 𝑎𝑟𝑟𝑒𝑠𝑡 𝑡𝑒𝑚𝑝𝑒𝑟𝑎𝑡𝑢𝑟𝑒

𝑛𝑜. 𝑜𝑓 𝑟𝑢𝑛𝑠=

(21 + 26 + 24 + 34 + 26 + 24)°𝐶

6= 25.83°𝐶

Theoretical Eutectic Composition

ln 𝜒𝐴 =Δ𝑓𝑢𝑠 𝐴𝐻

𝑅(

1

𝑇𝑚,𝐴−

1

𝑇𝑒)

ln 𝜒𝐴 =17863.88 𝐽/𝑚𝑜𝑙

8.314 𝐽/𝑚𝑜𝑙 ∙ 𝐾(

1

326.15−

1

𝑇𝑒)

ln 𝜒𝐵 = Δ𝑓𝑢𝑠 𝐵𝐻

𝑅(

1

𝑇𝑚,𝐵−

1

𝑇𝑒)

ln(1 − 𝜒𝐴) = 19305.48 𝐽/𝑚𝑜𝑙

8.314 𝐽/𝑚𝑜𝑙 ∙ 𝐾(

1

353.35−

1

𝑇𝑒)

1

𝑇𝑒= −ln 𝜒𝐴

8.314𝐽

𝑚𝑜𝑙∙ 𝐾

17863.88𝐽

𝑚𝑜𝑙

+1

326.15

1

𝑇𝑒= −ln(1 − 𝜒𝐴)

8.314𝐽

𝑚𝑜𝑙∙ 𝐾

19305.48𝐽

𝑚𝑜𝑙

+1

353.35

−ln 𝜒𝐴

8.314𝐽

𝑚𝑜𝑙∙ 𝐾

17863.88𝐽

𝑚𝑜𝑙

+1

326.15= −ln(1 − 𝜒𝐴)

8.314𝐽

𝑚𝑜𝑙∙ 𝐾

19305.48𝐽

𝑚𝑜𝑙

+1

353.35

−4.654084107 × 10−4 ln 𝜒𝐴 + 3.066073892 × 10−3

= −4.306549229 × 10−4 ln(1 − 𝜒𝐴) + 2.830055186 × 10−3

𝑓(𝑥) = −4.654084107 × 10−4 ln 𝜒𝐴 + 4.306549229 × 10−4 ln(1 − 𝜒𝐴) + 2.360187063 × 10−4 = 0

𝑓′(𝑥) =−4.654084107 × 10−4

𝜒𝐴−

4.306549229 × 10−4

(1 − 𝜒𝐴)= 0

𝑥𝑛+1 = 𝑥𝑛 −𝑓(𝑥)

𝑓′(𝑥); 𝑥1 = 0.5

𝜒𝐴 = 0.6419450191

𝜒𝐵 = 0.3580549809

ln 𝜒𝐴 =17863.88 𝐽/𝑚𝑜𝑙

8.314 𝐽/𝑚𝑜𝑙 ∙ 𝐾(

1

326.15−

1

𝑇𝑒)

ln 0.6419450191 =17863.88 𝐽/𝑚𝑜𝑙

8.314 𝐽/𝑚𝑜𝑙 ∙ 𝐾(

1

326.15−

1

𝑇𝑒)

𝑇𝑒 = 305.5891595 𝐾 − 273.15 = 32.43915948 °𝐶

Percent Error

Te

%𝑒𝑟𝑟𝑜𝑟 = 𝑎𝑐𝑡𝑢𝑎𝑙 − 𝑡ℎ𝑒𝑜𝑟𝑒𝑡𝑖𝑐𝑎𝑙

𝑡ℎ𝑒𝑜𝑟𝑒𝑡𝑖𝑐𝑎𝑙× 100 =

25.83 °C − 32.44 °C

32.44 °C× 100 = −20.37%

Eutectic Composition

𝜒𝐴 = 19.14% 𝜒𝐵 = 34.32%

0

10

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60

70

80

0 100 200 300 400 500 600 700

Series1

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0 100 200 300 400 500 600 700 800 900

Series1

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0 100 200 300 400 500 600 700

Series1

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0 50 100 150 200 250 300 350 400

Tem

pe

ratu

re (

°C)

Time (s)

Series1

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0 100 200 300 400 500 600

0

10

20

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0 50 100 150 200 250 300 350

Series1

Raw Data

Table 1.3 Data on mole fraction, break temperature and arrest temperature of each run

Run ΧB Tb, °C Ta, °C

6 0 40 -

7 0.2090141716 42 26

8 0.3062061820 47 24

5 0.3989950942 54 34

4 0.5703509791 46 24

3 0.7263318323 67 26

2 0.8690950608 65 21

1 1.00 68 -