# Water Phase Diagram: Mathematically Modeling and Validating

• DLawless
In summary, the online water phase diagrams often show a linear behavior for the solid-liquid boundary, with an extremely steep slope. This is modeled mathematically using the Clapeyron equation with constant values for ΔH and ΔV_m. However, when integrated, this does not result in a straight line. The almost straight line, suggesting P=kT for some very large negative k, is not consistent with the mathematical model. Many other diagrams also show a linear boundary, but it is not clear how this is achieved mathematically. Additionally, the non-linear curved parts at high pressure do not seem to follow the same equation.
DLawless
I notice that water phase diagrams provided online always seem to show a rather linear behaviour for the solid-liquid boundary (and an extremely steep slope).

How is this modeled mathematically? Say we use the Clapeyron equation with ΔH and ΔV_m being constant, as online example problems (meant for students) do. Integration with this yields a ln(T2/T1) for example--not the equation of a straight line. So where does the almost straight line, which suggests that P=kT for some very large negative k, come from? And how valid is it really to model ΔH and ΔV_m as constant, if this doesn't produce a line that looks much like the diagrams show?

It is not linear. Here is a phase diagram: It's complicated.
DLawless said:
Integration with this yields a ln(T2/T1) for example
For what, and what are T1 and T2?

The line is so steep because both a few hundred kPa is a low pressure in the context of water and ice. You see larger deviations at tens of MPa.

Many other diagrams show a line, e.g. https://uh.edu/~jbutler/physical/chapter6notes.html, https://scholar.harvard.edu/files/schwartz/files/9-phases.pdf. I saw the diagram you mentioned too.

I would like to know the mathematical model that leads to either of these. It doesn't appear to be constant ΔH and ΔV_m since neither the shape in the diagram you linked, nor a straight line, corresponds well to

$$P = P_0 + \frac{\Delta H}{\Delta V_m} \rm{ln} \frac{T}{T_0}$$

which is the result from integrating Clapeyron equation. (T0,P0) can be any known point, e.g. (273.15 K, 1 atm) for water solidus

Yet all the problems/examples one finds online seems to treat ΔH and ΔV_m as constant. If it is valid to do so I would like to know how the boundary can have the shape we see on the diagrams.

mfb said:
Locally (if T/T0 doesn't deviate too much from 1) a logarithm looks like a straight line.

Good point! That may explain the linear shape.

What about the non-linear curved bits at high pressure (that you can see on the diagram you linked originally, or the one here https://scholar.harvard.edu/files/schwartz/files/9-phases.pdf on p4)? They don't necessarily appear to follow the equation I gave...

Edit: actually it looks like there might be a cusp and another independent solidus?

The ##\Delta V## is tiny, and that accounts for the huge slope.

## 1. What is a water phase diagram?

A water phase diagram is a graphical representation of the physical states of water (solid, liquid, and gas) at different combinations of temperature and pressure. It shows the conditions at which water can exist as a solid, liquid, or gas, and the boundaries between these phases.

## 2. How is a water phase diagram mathematically modeled?

A water phase diagram is mathematically modeled using the Clausius-Clapeyron equation, which relates the temperature and pressure at which a substance changes state. This equation takes into account the enthalpy and entropy of the substance, as well as the temperature and pressure at which the phase change occurs.

## 3. What is the purpose of validating a water phase diagram?

The purpose of validating a water phase diagram is to ensure that the mathematical model accurately reflects the behavior of water in different physical states. This involves comparing the predicted phase boundaries and transition points to experimental data and making adjustments to the model as needed.

## 4. What factors can affect the accuracy of a water phase diagram?

Several factors can affect the accuracy of a water phase diagram, including the purity of the water sample, the presence of impurities, and the accuracy of the experimental measurements used to validate the model. Changes in atmospheric pressure and temperature can also affect the accuracy of the diagram.

## 5. How is a water phase diagram useful in scientific research?

A water phase diagram is useful in scientific research as it provides a visual representation of the behavior of water under different conditions. This can help researchers understand the properties of water and how they are affected by changes in temperature and pressure. The diagram can also be used to predict the behavior of other substances that have similar phase diagrams.

• Mechanics
Replies
4
Views
2K
• Classical Physics
Replies
45
Views
4K
• Chemistry
Replies
4
Views
3K
• Introductory Physics Homework Help
Replies
9
Views
3K
• Electrical Engineering
Replies
1
Views
1K
• Programming and Computer Science
Replies
1
Views
2K
• Engineering and Comp Sci Homework Help
Replies
4
Views
2K
• Mechanics
Replies
8
Views
7K
• Introductory Physics Homework Help
Replies
8
Views
2K
• Introductory Physics Homework Help
Replies
4
Views
2K