Chapter 11.6 - Phase Diagrams of Carbondioxide and Water

In the previous section we saw the basic details about phase diagram. Our discussions were based on fig.11.13. For convenience, that fig. is shown again below:

Fig.11.13

 

• Now we will see a solved example which will help us to get more information about phase diagrams. Later, we will see the phase diagram of water

Solved example 11.23
Fig.11.14 below shows the P-T diagram (phase diagram) of CO₂

Fig.11.14

Based on the diagram, answer the following questions:
(i) At what temperature and pressure can the solid, liquid and vapour phases of CO₂ co-exist in equilibrium ?
(ii) What is the effect of decrease of pressure on the fusion and boiling point of CO₂ ?
(iii) What are the critical temperature and pressure for CO₂ ? What is their
significance ?
(iv) Is CO₂ solid, liquid or gas at (a) –70 °C under 1 atm, (b) –60 °C under 10 atm,
(c) 15 °C under 56 atm ?
(v) CO₂ at 1 atm pressure and temperature – 60 °C is compressed isothermally.
Does it go through a liquid phase ?
(vi) What happens when CO₂ at 4 atm pressure is cooled from room temperature
at constant pressure ?
(vii) Describe qualitatively the changes in a given mass of solid CO₂ at 10 atm
pressure and temperature –65 °C as it is heated up to room temperature at
constant pressure.
(viii) CO₂ is heated to a temperature 70 °C and compressed isothermally. What
changes in its properties do you expect to observe ?
Solution:
Part (i):
• From the fig.11.14, we see that, the coordinates of the triple point are: (-56.6, 5.11)
    ♦ So the required temperature is: -56.6 °C
    ♦ Required pressure is: 5.11 atm
Part (ii):
1. First we will consider the fusion point
• This point will lie on the yellow curve
• Consider the ‘⨯’ mark on the yellow line in the previous fig.11.13
• When we decrease the pressure, we move downwards along the vertical double headed arrow
    ♦ So we fall out of the yellow line
• But for the fusion point, we must be exactly on the yellow curve
• So, at a lower pressure, to get back on the yellow curve, we must move towards the left
    ♦ That means, we have to reduce the temperature
• Thus, when pressure is reduced, the fusion point (temperature at which fusion takes place) of CO₂ is reduced
2 Next, we will consider the boiling point
• This point will lie on the green curve
• Consider the ‘⨯’ mark on the green line in the previous fig.11.13
• When we decrease the pressure, we move downwards along the vertical double headed arrow
    ♦ So we fall out of the green line
• But for the boiling point, we must be exactly on the green curve
• So, at a lower pressure, to get back on the green curve, we must move towards the left
    ♦ That means, we have to reduce the temperature
• Thus, when pressure is reduced, the boiling point (temperature at which boiling takes place) of CO₂ is reduced
Part (iii):
1.From the fig.11.14, we see that, the coordinates of the critical point are: (31.1,73)
    ♦ So the required temperature is: 31.1 ^_oC
    ♦ Required pressure is: 73 atm
2. Imagine that, keeping pressure constant at 73 atm, the temperature of CO₂ is increased to a value just above 31.1 ^_oC
    ♦ Then the critical point will move horizontally towards the right
    ♦ That means, the critical point will move out of the green curve
    ♦ That means, the critical point is out of the blue (liquid) region
• So we can write:
If the temperature of CO₂ is raised above 31.1 ^_oC, it will not liquefy even if we apply very high pressures
Part (iv):
(a) The given point is: (-70, 1)
• The point (-78.5, -1) is already marked in fig,11.14
    ♦ The given point (-70, -1) has the same y-coordinate
    ♦ So the given point will lie on a horizontal line through (-78.5, -1)
• Now, -70 is greater than -78.5. So (-70, 1) will lie on the right side of (78.5, -1)
    ♦ That means, (-70, -1) will lie in the vapour region
(b) The given point is (-60, 10)
• The points (-78.5, 1) and (56.6, 5.11) are already marked in fig.11.14
    ♦ -60 is greater than -78.5, but less than -56.6
    ♦ So the given point will lie on a vertical line between (-78.5, 1) and (56.6, 5.11)
• That means, the given point will either be a solid or vapour
• The y-coordinate of the given point is 10
    ♦ This is greater than 5.11
• So the point will lie in the solid region
(c) The given point is (15, 56)
• The point (20, 56) is already marked in fig,11.14
    ♦ The given point (15, 56) has the same y-coordinate
    ♦ So the given point will lie on a horizontal line through (20, 56)
• Now, 15 is lesser than 20. So (15, 56) will lie on the left side of (20, 56)
    ♦ Also, 15 is close to 20
• That means, (20, 56) will lie in the liquid region
Part (v):
• The given point is (-60, 1)
• ‘Isothermal’ means same temperature
• So, when CO₂ is compressed isothermally, the pressure will increase but the temperature will remain constant
    ♦ So it will move along a vertical line through (-60,1)
• This vertical line will lie in between (-78.5, 1) and (-56.6, 5.11)
• When the sample is compresses, it’s pressure increases above 1 atm
• From (-78.5,1), the position of 1 atm is available to us. It lies on the red curve
• So it is clear that, when pressure rises above 1 atm, the sample will be in the solid region. So it does not go through a liquid phase
Part (vi):
• The given point is (room temperature, 4)
• It is cooled at constant pressure. So it will pass along a horizontal line through (room temperature, 4)
• The y-coordinate of the triple point is 5.11
    ♦ So the horizontal line lies below the triple point
    ♦ So, when the sample is cooled, it passes from vapour region to solid region
• That means, deposition of CO₂ will take place
Part (vii):
• The given point is (-65, 10)
• It is heated at constant pressure. So it will pass along a horizontal line through (-65, 10)
• The y-coordinate of the triple point is 5.11
    ♦ So the horizontal line lies above the triple point
• The x-coordinate of the triple point is -56.6
    ♦ The given x-coordinate '-65' is less than -56.6
    ♦ So the initial position (-65,10) is in the solid region
• So, when the sample is heated, it first passes from solid to liquid. Then it passes from liquid to vapour
Part (viii):
• The given point can be written as (70, P)
• It is compressed isothermally. So it will pass along a vertical line through (70, P)
• The x-coordinate of the critical point is 31.1
    ♦ So the vertical line lies to the right of the critical point
• At such a high temperature, the CO₂ gas will not liquefy even if we apply very high pressures

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Now we will see the P-T diagram of water. It is shown in fig.11.15 below:

Fig.11.15

An explanation can be written in 8 steps:
1. The triple point of water is (0.01, 0.006)
• The temperature of 0.01 _^oC is very close to our familiar freezing point of 0 _^oC
• But the pressure of 0.006 atm is very much less than our familiar standard atmospheric pressure of 1 atm
2. It may be noted that, (0.01, 0.006) is the triple point. Solid ice, liquid water and gaseous water vapour will be present at that point
• At (0, 1), only solid ice and liquid water will be present
3. For most substances, the yellow curve starts from the triple point and slopes upwards towards the right. It is a positive slope
    ♦ That means, as temperature increases, pressure also increases
• But for water, the yellow curve starts from the triple point and slopes upwards towards the left. It is a negative slope
    ♦ That means, as temperature increases, pressure decreases
4. Let us see the reason for the negative slope. It can be written in 5 steps:
(i) Consider a sample of water at 0 _^oC
    ♦ If we heat it, it’s volume will decrease
(ii) This is opposite to what we expect. Because, substances usually increase in volume when heated
(We saw this anomalous behaviour of water in a previous section. Fig.11.5 of section 11.2)
(iii) Normally, if we heat a substance in a closed container, it’s volume will increase and so pressure will also increase
(iv) But if it is water at 0 _^oC, volume will decrease and so pressure will also decrease
(v) That means, with increase in temperature, we are having a decrease in pressure
• That is the reason for the negative slope of the yellow curve
5. Now, consider the freezing point of water
• We know that, at the normal atmospheric pressure of 1 atm, the freezing point is 0 _^oC
    ♦ So it is marked as (0,1)
    ♦ Note that, the boiling point (100,1) will fall at the same horizontal level
6. Keeping the temperature of ice at 0 _^oC, increase the pressure slightly
• We will then be moving vertically upwards from (0,1)
• When we move vertically upwards from (0,1), we will fall in the liquid region
• That means, even a slight increase in pressure, will cause the ice to liquefy
7. Ice skating is possible because:
• The pressure due to the weight of the skater, cause the upper layer of ice to change into a thin layer of water
• This thin layer of water acts as a lubricant
8. As soon as the pressure is removed, the ice refreezes again. This phenomenon of refreezing is called regelation
9. Regelation is possible due to the negative slope of the yellow curve
• Let us see what would happen if it was a positive slope
    ♦ Keeping the temperature of ice at 0 _^oC, increase the pressure
    ♦ We will then be moving vertically upwards from (0,1)
    ♦ When we move vertically upwards from (0,1), we will fall in the solid region
    ♦ That means: If the slope is positive, 'increase in pressure' will not cause the ice to liquefy

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In the next section, we will see heat transfer by conduction




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Chapter 11.5 - The Phase Diagram

In the previous section we saw change of phase. In this section we will see phase diagram and triple point

First let us see the relation between pressure and temperature. It can be written in 7 steps:
1. We have seen that, when external pressure on the surface of a liquid is increased, it becomes difficult for that liquid to boil
• That means, when the external pressure is increased, it becomes difficult to change from liquid state to gaseous state
• We can say that, external pressure tries to prevent the liquid from changing into a gas
2. We know that: Liquid is denser than gas
• So we can say that, external pressure tries to compress the body into a denser state
• If pressure is increased further, the liquid will even become a solid
3. But besides pressure, we have to consider another factor also. It is: temperature
• Temperature acts in the opposite sense
• When temperature is increased, it becomes difficult for a liquid to stay in the liquid state. It will be forced to change into gaseous state
• Similarly, when temperature is increased, it becomes difficult for a solid to stay in the solid state. It will be forced to change into liquid state
4. So we can write two points:
(i) When pressure increases, an increase in density occurs in the direction:
Gas → Liquid → Solid
(ii) When temperature increases, a decrease in density occurs in the direction:
Solid → Liquid → Gas
5. The reverse can also be written:
(i) When pressure decreases, a decrease in density occurs in the direction:
Solid → Liquid → Gas
(ii) When temperature decreases, an increase in density occurs in the direction:
Gas → Liquid → Solid
    ♦ For example, we obtain solid ice by decreasing the temperature of liquid water
6. We get an interesting result from 4(i) and 5(ii):
• If we can obtain sufficiently high pressure, and sufficiently low temperature, we can change gases into solids.
    ♦ For example, scientists have succeeded in making solid hydrogen in the lab
          ✰ Note that, the end result in both 4(i) and 5(ii) is 'solid'
7. So pressure and temperature has a somewhat inverse relation with each other:
    ♦ High pressure and low temperature gives solid state
    ♦ Low pressure and high temperature gives gaseous state
    ♦ Intermediate values of pressure and temperature gives liquid state
• We can plot a graph with pressure along the y-axis and temperature along the x-axis
• Such a graph is known as P-T diagram or phase diagram

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Before discussing about the P-T diagram, we have to first become familiar with the chart shown in fig.11.12 below:

Fig.11.12

• The chart shows that:
    ♦ A change from solid state to liquid state is called melting
    ♦ A change from liquid state to solid state is called freezing
    ♦ A change from liquid state to gaseous state is called vaporization
    ♦ So on . . .
• We will need to mention these terms when we discuss about P-T diagram

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• Now we will see the details about P-T diagram
• Every substance has it’s own unique P-T diagram
• Scientists have already prepared them for various substances
    ♦ All we need now, is to learn the common features of those diagrams
    ♦ While we discuss those common features, we will learn how a P-T diagram is prepared
• The features can be written in 13 steps
1. Fig.11.13 below shows a P-T diagram
    ♦ Pressure is plotted along the y-axis
    ♦ Temperature is plotted along the x-axis

Fig.11.13

2. When the x and y axes are drawn, we get the x-y plane
    ♦ In our present case, x-y plane is the P-T plane
• In the fig.11.13 above, the P-T plane is divided into three regions: orange, blue and gray
3. In the P-T plane, there are infinite number of points
• Each of those points will be in the form: (T.P)
• Pick any one of those points from the plane. Let it be (Ti, Pi)
• We can write three properties:
(i) If the point is from the orange region:
    ♦ Apply a pressure of Pi on a sample of that substance
    ♦ Bring the temperature of that sample to Ti
    ♦ Then the sample will change into solid state
(ii) If the point is from the blue region:
    ♦ Apply a pressure of Pi on a sample of that substance
    ♦ Bring the temperature of that sample to Ti
    ♦ Then the sample will change into liquid state
(iii) If the point is from the gray region:
    ♦ Apply a pressure of Pi on a sample of that substance
    ♦ Bring the temperature of that sample to Ti
    ♦ Then the sample will change into gaseous state
4. Note that, the solid region (orange) is towards the left of the graph
    ♦ Here, temperatures will be low
    ♦ We know that, at low temperatures, substances tend to solidify
• In this way, we can characterize the blue and gray regions also
• Now we know the significance of the three regions in the P-T diagram
5. Next, we will see the three curves: red, yellow and green
    ♦ The yellow curve separates the solid and liquid regions
    ♦ The green curve separates the liquid and gaseous regions
    ♦ The red curve separates the solid and gaseous regions
6. Consider the yellow curve
We can write a description in 7 steps:
(i) There are infinite number of points on the yellow curve
• Take any one point from them. Let it be (Ti,Pi). It is marked with a ‘⨯’
    ♦ Apply a pressure of Pi on a sample of that substance
    ♦ Bring the temperature of that sample to Ti
(ii) Then we can write:
    ♦ If the sample is initially in the solid state, it will begin to melt
    ♦ If the sample is initially in the liquid state, it will begin to freeze
(iii) This is because, the yellow curve is the transition between solid and liquid
• Along this curve, the sample is partially solid and partially liquid
(iv) keeping temperature constant at Ti, let us increase the pressure
• This is indicated by the vertical double headed arrow
    ♦ The pressure moves upwards along that arrow
    ♦ Then the sample is no longer ‘melting/freezing’
    ♦ All the liquid portion will freeze
    ♦ Thus It will enter the orange region and become a complete solid
• This is what we would expect because, increasing pressure gives denser substances
(v) keeping temperature constant at Ti, let us decrease the pressure
    ♦ Now the pressure moves downwards along the vertical double headed arrow
    ♦ Then the sample is no longer ‘melting/freezing’
    ♦ All the solid portion will melt
    ♦ Thus it will enter the blue region and become a complete liquid
• This is what we would expect because, decreasing pressure gives less denser substances
(vi) keeping pressure constant at Pi, let us increase the temperature
• This is indicated by the horizontal double headed arrow
    ♦ The temperature moves towards the right along that arrow
    ♦ Then the sample is no longer ‘melting/freezing’
    ♦ All the solid portion will melt
    ♦ Thus It will enter the blue region and become a complete liquid
• This is what we would expect because, increasing temperature gives less denser substances
(vii) keeping temperature constant at Ti, let us increase the pressure
    ♦ Now the pressure moves towards the left along the double headed arrow
    ♦ Then the sample is no longer ‘melting/freezing’
    ♦ All the liquid portion will freeze
    ♦ Thus it will enter the orange region and become a complete solid
• This is what we would expect because, increasing pressure gives denser substances
7. So the yellow curve is a transition between solid and liquid
We can write similar steps for the green curve:
(i) Consider any convenient point on the green curve. Let it be (Ti,Pi). It is marked with a ‘⨯’
    ♦ Apply a pressure of Pi on a sample of that substance
    ♦ Bring the temperature of that sample to Ti
• A portion of the sample will be in the liquid state
• The remaining portion will be in the gaseous state
(ii) Keeping temperature constant, increase the pressure
    ♦ This is an upward motion along a vertical double headed arrow
    ♦ The vapor portion will condense and the sample will become a complete liquid
    ♦ The sample will enter the blue region
(iii) Keeping temperature constant, decrease the pressure
    ♦ This is a downward motion along the vertical double headed arrow
    ♦ The liquid portion will vaporize and the sample will become a complete vapor
    ♦ The sample will to enter the gray region
(iv) Keeping pressure constant, increase the temperature
    ♦ This is a rightward motion along a horizontal double headed arrow
    ♦ The liquid portion will vaporize and the sample will become a complete vapor
    ♦ The sample will enter the gray region
(v) Keeping pressure constant, decrease the temperature
    ♦ This is a leftward motion along the horizontal double headed arrow
    ♦ The vapor portion will condense and the sample will become a complete liquid
    ♦ The sample will enter the blue region
8. We can write similar steps for the red curve:
(i) Consider any convenient point on the red curve. Let it be (Ti,Pi). It is marked with a ‘⨯’
    ♦ Apply a pressure of Pi on a sample of that substance
    ♦ Bring the temperature of that sample to Ti
• A portion of the sample will be in the solid state
• The remaining portion will be in the gaseous state
(ii) Keeping temperature constant, increase the pressure
    ♦ This is an upward motion along a vertical double headed arrow
    ♦ The vapor portion will deposit and the sample will become a complete solid        ♦ The sample will enter the orange region
(iii) Keeping temperature constant, decrease the pressure
    ♦ This is a downward motion along the vertical double headed arrow
    ♦ The solid portion will sublime and the sample will become a complete vapor
    ♦ The sample will to enter the gray region
(iv) Keeping pressure constant, increase the temperature
    ♦ This is a rightward motion along a horizontal double headed arrow
    ♦ The solid portion will sublime and the sample will become a complete vapor
    ♦ The sample will enter the gray region
(v) Keeping pressure constant, decrease the temperature
    ♦ This is a leftward motion along the horizontal double headed arrow
    ♦ The vapor portion will deposit and the sample will become a complete solid
    ♦ The sample will enter the orange region
9. So the significance of the three curves is now clear. It can be written in 3 steps:
(i) Along the yellow curve:
    ♦ A portion of the sample will be in the solid state
    ♦ The remaining portion will be in the liquid state
■ This curve is called fusion curve
(ii) Along the green curve:
    ♦ A portion of the sample will be in the liquid state
    ♦ The remaining portion will be in the gaseous state
■ This curve is called vaporization curve
(ii) Along the red curve:
    ♦ A portion of the sample will be in the solid state
    ♦ The remaining portion will be in the gaseous state
■ This curve is called sublimation curve
10. So, at the point of intersection of the three curves:
    ♦ A portion of the sample will be in the solid state
    ♦ Another portion of the sample will be in the liquid state
    ♦ The remaining portion will be in the gaseous state
■ This point of intersection of the three curves is called triple point
11. Imagine a horizontal line through the triple point
• Above that horizontal line, orange, blue and gray regions are present
• Below that horizontal line, only orange and gray regions are present
    ♦ There is no blue region
■ So it is clear that:
If the pressure of a sample is below the ‘pressure at triple point’, then that sample cannot be in the liquid state
12. Imagine a vertical line through the triple point
• To the right of that vertical line, orange, blue and gray regions are present
• To left of that vertical line, only orange and gray regions are present
    ♦ There is no blue region
■ So it is clear that:
If the temperature of a sample is below the ‘temperature at triple point’, then that sample cannot be in the liquid state
13. Another important point is the critical point
• This point is at upper portion and towards the right side of the graph
    ♦ Since it is at the upper portion, pressure will be very high
    ♦ Since it is towards the right side, temperature will also be very high
• But even if the pressure is very high, substances cannot exist as solid
    ♦ This is because of the very high temperature
• So at the critical point, portion of the sample will be in the liquid state
    ♦ The remaining portion will be in the gaseous state
• If the temperature or pressure is increased beyond the critical point, the liquid and gaseous portions will become indistinguishable from each other
• We will learn about the applications of critical point in higher classes

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Some solved examples will help us to get a good understanding about phase diagrams. We will see them in the next section

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