Gibbs Phase Rule (Examples)

Summary

The video script discusses the Gibbs Phase Rule, which determines the degrees of freedom in a system by subtracting the number of coexisting phases from the number of components and adding two. Examples include single-component systems, multi-component solutions like air and carbonated water, solid solutions like brass, and complex systems with multiple phases and components. The rule is shown to be applicable regardless of system complexity, illustrating how it dictates the number of variables that can be independently controlled.

Takeaways

• 📚 The Gibbs Phase Rule states that the number of degrees of freedom in a system is equal to the number of components minus the number of coexisting phases plus two.
• 🔍 In a single-component system, the degrees of freedom are calculated as 1 (for the component) + 2 (from the rule), resulting in 3 degrees of freedom.
• 🌡️ For a two-component single-phase system, like a gaseous mixture of nitrogen and oxygen, the degrees of freedom are also three, allowing specification of temperature, pressure, and composition.
• 💧 In the case of carbonated water, a two-component system with liquid and gas phases, the degrees of freedom are reduced to two, as the pressure of the vapor phase components is determined by phase equilibrium.
• 🔩 Brass, a solid mixture of copper and zinc, has three degrees of freedom, allowing for the specification of the mole fraction of copper, temperature, and pressure, with the mole fraction of zinc being determined by the first.
• 🍬 A saturated sucrose solution has two degrees of freedom, as the concentration of sugar is fixed at the saturation point, leaving only temperature and pressure as variables.
• 🧪 For a system with three components in three phases, such as a liquid solution of water and propanol with a dissolved solute like sugar or salt, the degrees of freedom are two, despite the complexity of the system.
• 🌡️💦 The phase rule applies to both simple and complex systems, indicating that the number of variables that can be independently controlled is limited by the components and phases present.
• 🔄 The rule helps in understanding that once certain variables are set, others will be determined by the equilibrium conditions of the system.
• 📉 In a saturated solution, increasing the concentration of the solute beyond the saturation point will result in precipitation, and decreasing it will cause dissolution to maintain equilibrium.
• 🌤️ The phase rule is a fundamental principle in thermodynamics that helps predict the behavior of systems at equilibrium, regardless of the number of components or phases involved.

Q & A

What does the Gibbs Phase Rule state?

The Gibbs Phase Rule states that the number of degrees of freedom in a system is equal to the number of components minus the number of phases coexisting at equilibrium, plus two.

How many degrees of freedom are there in a single-component system according to the Gibbs Phase Rule?

In a single-component system, there are three degrees of freedom. This is calculated as the number of components (1) minus the number of phases (assumed to be 1), plus two.

What is an example of a two-component single-phase system and its degrees of freedom?

An example of a two-component single-phase system is a mixture of nitrogen and oxygen, like air. The degrees of freedom for this system is three, calculated as the number of components (2) minus the number of phases (1), plus two.

How does the Gibbs Phase Rule apply to a system with two components in two different phases?

For a system with two components in two different phases, such as a carbonated water system with liquid water and gaseous CO2, the degrees of freedom is two. This is calculated as the number of components (2) minus the number of phases (2), plus two.

What is an example of a solid phase system with two components and its degrees of freedom?

An example of a solid phase system with two components is brass, which is a mixture of copper and zinc. The degrees of freedom for this system is three, calculated as the number of components (2) minus the number of phases (1, the solid phase), plus two.

What happens to the degrees of freedom when a solution becomes saturated with a solute?

When a solution becomes saturated with a solute, the degrees of freedom decrease by one. For a two-component system with two phases coexisting (like a saturated sugar solution), the degrees of freedom are two, as you can specify temperature and pressure but not the concentration of the solute independently.

Why can't you specify more than two degrees of freedom in a saturated solution with two components and two phases?

In a saturated solution with two components and two phases, you can only specify two degrees of freedom because the concentration of the solute is fixed at the saturation point. Any attempt to change the concentration will result in precipitation or dissolution to maintain equilibrium.

What is the degrees of freedom for a system with three components in three phases?

For a system with three components in three phases, the degrees of freedom is two. This is calculated as the number of components (3) minus the number of phases (3), plus two.

How does the presence of a non-volatile solute in a two-component two-phase system affect the degrees of freedom?

The presence of a non-volatile solute in a two-component two-phase system does not change the degrees of freedom. The system still has two degrees of freedom, as the non-volatile solute will precipitate into a solid phase and does not affect the vapor pressure equilibrium of the volatile components.

Can the Gibbs Phase Rule be applied to complex systems with multiple components and phases?

Yes, the Gibbs Phase Rule can be applied to any system, no matter how complex, with multiple components and phases. It will accurately determine the number of independent variables that can be controlled in the system.

What is the significance of the Gibbs Phase Rule in understanding phase equilibrium?

The Gibbs Phase Rule is significant in understanding phase equilibrium as it provides a quantitative relationship between the number of components, phases, and degrees of freedom in a system. It helps predict the behavior of a system when variables such as temperature, pressure, and composition are altered.

Outlines

00:00

🔍 Introduction to Gibbs Phase Rule

The first paragraph introduces the Gibbs Phase Rule, which is a fundamental concept in thermodynamics. It explains that the number of degrees of freedom in a system is determined by the formula: the number of components minus the number of coexisting phases plus two. The paragraph provides examples to illustrate this rule, including single-component systems and multi-component single-phase systems like air. It also discusses the case of carbonated water, a two-component two-phase system, and how the degrees of freedom are limited once the system reaches a certain state of equilibrium.

05:01

📚 Applying the Gibbs Phase Rule to Various Systems

This paragraph delves deeper into applying the Gibbs Phase Rule to different types of systems, starting with a solid phase example using brass, a mixture of copper and zinc. It then moves on to a single-phase solution of sucrose in water, explaining how the degrees of freedom change when the solution becomes saturated and precipitates the solute. The discussion includes the loss of a degree of freedom when the system reaches saturation, and the inability to independently control all variables in such a state.

10:04

🌐 Complex Systems and the Gibbs Phase Rule

The final paragraph explores the application of the Gibbs Phase Rule to more complex systems, including a three-component three-phase system with a liquid, vapor, and solid phase. It uses the example of a solution of water and propanol with a dissolved solute like sugar or salt, which can precipitate in the solid phase. The paragraph emphasizes that despite the complexity of the system, the Gibbs Phase Rule accurately predicts the number of degrees of freedom, illustrating that the rule is universally applicable regardless of the system's composition or phase distribution.

Keywords

💡Gibbs Phase Rule

The Gibbs Phase Rule is a fundamental principle in thermodynamics that relates the number of components in a system to the degrees of freedom available for the system to exist at equilibrium. It is defined as the number of degrees of freedom (F) being equal to the number of components (C) minus the number of phases (P) plus two, expressed as F = C - P + 2. In the video, this rule is used to determine the flexibility in specifying variables for various systems, such as single-component systems and multi-component solutions.

💡Degrees of Freedom

In the context of the video, degrees of freedom refer to the number of independent variables that can be specified for a system at equilibrium without violating the constraints imposed by the phase rule. The script uses this concept to explain the limitations on the variables that can be controlled in different types of systems, such as a single-component system or a multi-component solution.

💡Components

Components in the script refer to the distinct chemical substances that make up a system. The number of components is a key factor in applying the Gibbs Phase Rule, as it directly influences the number of degrees of freedom. For example, a system with nitrogen and oxygen (air) has two components, which affects the calculation of its degrees of freedom.

💡Phases

Phases in the video denote the distinct states or regions of matter in a system, such as solid, liquid, or gas. The coexistence of different phases in a system is crucial for applying the Gibbs Phase Rule, as it determines the equilibrium conditions and the number of degrees of freedom. The script discusses various phase coexistence scenarios, like liquid and vapor in a carbonated water system.

💡Equilibrium

Equilibrium in the script refers to a state where a system has reached a balance such that there is no net change in its properties. The Gibbs Phase Rule is specifically concerned with conditions at equilibrium, as it is only then that the rule can be applied to determine the degrees of freedom. The video uses equilibrium to discuss how variables are interdependent in systems like brass or a saturated sugar solution.

💡Mole Fraction

Mole fraction is a measure of the amount of a particular component in a mixture, expressed as the ratio of the number of moles of that component to the total number of moles of all components. The video script uses mole fraction as an example of a variable that can be specified in multi-component systems, such as in a brass mixture or a sugar-water solution.

💡Vapor Pressure

Vapor pressure is the pressure exerted by a vapor in equilibrium with its condensed phases at a given temperature in a closed system. In the context of the video, vapor pressure is discussed in relation to the phase equilibrium of volatile components like CO2 and H2O in a carbonated water system, determining the conditions under which a substance will evaporate or condense.

💡Saturation

Saturation in the script refers to the point at which a solution has reached the maximum concentration of solute that can be dissolved at a given temperature and pressure. The concept is used to illustrate the loss of a degree of freedom in a system, as the concentration of the solute becomes fixed at the saturation point, as seen in the example of a saturated sucrose solution.

💡Non-Volatile Solute

A non-volatile solute, as mentioned in the script, is a substance that does not readily evaporate or transition into the vapor phase under the conditions of the system. The script uses this term to describe a solute like salt, which remains in the liquid and solid phases but does not contribute to the vapor phase in a three-component system.

💡Thermodynamic Variables

Thermodynamic variables in the video are properties of a system that can be quantitatively described, such as temperature, pressure, and concentration. The script discusses how the Gibbs Phase Rule limits the number of these variables that can be independently specified at equilibrium, especially in systems with multiple phases and components.

Highlights

Introduction to the Gibbs Phase Rule and its significance in determining the degrees of freedom in a system.

Explanation of the formula for the Gibbs Phase Rule: F = C - P + 2, where F is degrees of freedom, C is components, and P is phases.

Application of the Gibbs Phase Rule to a single component system, resulting in three degrees of freedom.

Example of a two-component single phase system, such as air, with three degrees of freedom.

Discussion on the degrees of freedom in a carbonated water system, illustrating the rule's application to two-phase systems.

Exploring degrees of freedom in a solid phase system, like brass, which has three degrees of freedom.

Analysis of a single-phase solution, such as sugar water, and its three degrees of freedom.

Transition from a single-phase to a two-phase system with a saturated sucrose solution and the resulting loss of a degree of freedom.

Clarification on why the concentration of sugar in a saturated solution cannot be independently specified.

Complex example involving a three-component system with three phases, demonstrating the rule's versatility.

Explanation of the limited degrees of freedom in a three-component, three-phase system, despite its complexity.

Illustration of how the Gibbs Phase Rule applies to systems with varying components and phases, emphasizing its universality.

Final summary emphasizing that the Gibbs Phase Rule dictates the number of variables that can be controlled in any system.

Transcripts

all right so the gibbs phase rule

tells us that the number of degrees of

freedom we have the flexibility to

choose is equal to the number of

components of a solution

of a system minus the number of phases

coexisting at equilibrium in that system

plus two so let's work a few examples

and make sure

that makes sense to us we can first do a

couple of examples where we already know

the answer

the

case of a single component system we

just have one component

so c equals 1 then if i just plug x

equals 1 into this equation

1 plus 2 is equal to 3 minus 5

and that's the same result we've gotten

previously for the number of degrees of

freedom

accessible to us in a single component

system

so the gibbs phase rule is a more

generalized version

of the rule we've used previously for

single phases

if we want to talk about multiple

component solutions or systems

the two examples we considered by hand

in the previous lecture

the first of those was a gaseous system

with a composition

something like air a mixture of nitrogen

and oxygen so that was a

two component single phase system

the number of degrees of freedom is two

components

minus one phase plus two

so two minus one plus two that equals

three and that's

the result we convinced ourselves uh

was reasonable for that system for

example

we could specify the temperature and the

pressure and the composition

of that mixture but once we've specified

three variables i can't also specify the

mole fraction of oxygen

per the other example we thought about

in the previous lecture was carbonated

water

a mixture of co2 and h2o

in the liquid and gas phases

coexisting so the picture we drew in

that case was

liquid water dissolved co2

water and co2 both in the gas phase

that's again a two component system co2

and h2o

two phases coexisting with each other

so the number of degrees of freedom

components

minus phases plus 2

that gives me 2. so we have only two

variables we can specify

again that matches what we convinced

ourselves was the case

in that example i can dissolve a certain

concentration

of co2 i can set the temperature to

whatever i want but then the pressures

of the vapor phase components will be

determined

by the phase equilibrium so for example

i could set

temperature and concentration of co2

i could set i could put the

system under whatever pressure i want i

can choose the temperature i can choose

the pressure but once i've determined

the pressure

if the pressure in the vapor phase is

higher

than the vapor pressure of co2 it will

dissolve into the liquid if the pressure

is higher than the vapor pressure

of h2o it will dissolve it will condense

down into the liquid phase

so i can choose

two variables but once i've chosen those

two the other ones will be

determined for me to work a few examples

we haven't considered yet

let's do one in the solid phase

let's take a mixture of two compounds in

the solid phase so for example

brass is a mixture of copper and zinc

how many degrees of freedom i'm am i

allowed to specify

for a chunk of brass so that's

two components copper and zinc

one phase i'm just talking about the

solid so there's only a single phase

so components minus

phases plus two two minus 1 plus 2 is

again equal to 3 that's the same math as

in

this situation so is that reasonable can

i think of three different variables

that i could specify

for a sample of solid brass

i can i can specify

the mole fraction of copper

in that sample and then i can take that

sample and i can

heat it to whatever temperature i want i

can set the i can put it under some

amount of pressure

there's no contradiction between doing

all those three things at the same time

but again i couldn't also independently

set the mole fraction of the other

component once i choose

the mole fraction of copper mole

fraction of zinc is determined for me

let's take uh

as a multi-phase system instead of solid

instead of liquid and vapor phase let's

do one

actually first let's do

a solution with only a single phase so

let's

say

pure liquid so there's no vapor in this

case

just a container canning liquid and

let's do

i want to eventually consider a solution

that we could

over saturate so let's take a solute

like sugar

sucrose i'll dissolve some sugar in

water to make a sugar water solution

so that's two components

single phase again like every time we've

considered two components single phase

two minus one plus two

is going to work out to three different

variables that we could independently

specify

and again that makes sense in this case

i can choose a composition variable i

can choose

the concentration of sucrose i can make

a one molar

solution of sucrose i can make a half

molar solution of sucrose

i can place that sample at any

temperature i want

and any pressure i want within a certain

range of

possible values so that's very much like

the single phase systems we've

considered in the gas phase or the

the solid phase but now

let's take that the concentration of

that sucrose solution to a point where

it's saturated

i'll keep dissolving

sucrose in my water until it's so

concentrated that the sucrose begins to

precipitate

so i'm recording this in south carolina

where we like our sweet tea so

the way to make sweet tea is there's all

so much water

that it precipitates and sits on the

bottom of the solution so that saturated

solution now i've got

two components and two different phases

in coexistence

water and sucrose are the two components

the two phases are the liquid phase and

the solid phase

so now the number of degrees of freedom

two minus two plus two

that works out to two degrees of freedom

that's a different answer than i had for

this saturated solution this solution

below the saturation concentration so

which

variable have i lost why does it make

sense that i can no longer specify all

three of these

concentrate these thermodynamic

variables at the same time

i could specify

the temperature and the pressure i can

certainly take my sugar water solution

and heat it up or cool it down make it

any range of temperatures

i can put it under some pressure there's

nothing to stop me from doing that

but what i can't specify any longer is

the concentration of sugar

so i can only specify temperature and

pressure if i try to

increase the concentration of sugar to

even more concentrated concentrations if

i add more sugar

sugar into this solution then what's

going to happen is it won't stay in

solution

it will just precipitate out the keeping

the concentration of the solution at the

saturation concentration

likewise if i try to reduce the

concentration

up here in the liquid phase by removing

some sucrose molecules from the liquid

phase

because of the equilibrium between these

two phases some solid molecules will

just dissolve

and replace the moltens i've taken out

of the liquid solution so

the the sucrose concentration is fixed

at the

saturation concentration at this

particular temperature and pressure

so the gibbs phase rule is correctly

telling us that we've lost a degree of

freedom

in this case and as one final example

just to take something even more

complicated to show you that

you don't need just simple two-phase

cases you don't need two component

solutions let's

make an example

that's even more complicated let's say

we have a

liquid in equilibrium with vapor let's

make it a two component solution

so i've got who knows water and

propanol in this liquid solution

both of those are volatile solvents and

they'll have vapor

up in the gas phase but now into this

two component two phase solution let's

also dissolve a solute so let's dissolve

a third component maybe sugar or salt or

something else

that's in the liquid phase if it's salt

it's a non-volatile solute so there

won't be any up in the vapor phase

but it will if i'm at saturation

conditions precipitate down into the

solid phase

so now i've got three components

propanol and salt or or three different

components

in three different phases vapor phase

liquid phase solid phase

so that tells me the number of degrees

of freedom in this case is three

components

minus three phases plus two i still

only despite the complicated uh setup of

the system i've only got

two degrees of freedom that i'm allowed

to specify according to

the gibbs phase rule so does that make

sense

why can't i specify more than just two

degrees of freedom

as an example i can

dissolve i can certainly set the mole

fraction of compound on a

in the liquid phase so i can make a 50

50

mixture of the two solvents if i want to

i can heat or cool that system down to

whatever temperature i'm interested in

but i can't choose the mole fraction of

c

since i'm at saturation then this is

going to be the saturation concentration

i can't choose the amount of a in the

vapor phase at this particular

temperature the vapor pressure of a is

some number

at this particular temperature the vapor

pressure of b is some number so those

two pressures are going to add to some

total pressure

in the gas phase so i can't choose the

pressure

to apply on the gas phase if i try to

make it some different value

then either more molecules of a or b

will evaporate to replace

the to raise the pressure or

a and b will condense to lower the

pressure so once i've chosen those two

degrees of freedom or any two degrees of

freedom that i try to choose

independently all the other ones will be

determined so it doesn't matter how

complicated the system is doesn't matter

how many components how many phases

doesn't matter if some components

contain all the phases

some contain only one phase some contain

only

some contain only some one of the

components some phases may contain a

subset of the components

as complicated as a system as you can

construct the gibbs phase will tell you

how many different variables you can try

to control at the same time