# Hyperbola Transformations in Clapeyron-Mendeleev Coordinates

• Iustin Ouatu
In summary, the conversation discusses the representation of an isotherm transformation of an ideal gas in Clapeyron-Mendeleev coordinates and the equation of a hyperbola. The participants also discuss rotating and shifting the graph of a hyperbola, as well as defining a new system of coordinates.
Iustin Ouatu
Back in 10th degree, I have learned that in Clapeyron-Mendeleev coordinates ( eq: p-V) , an Isotherm transformation of an ideal gas ( with constant mass throughout the transformation ) is represented with an arc of an hyperbola. Now, I have learned that hyperbola equation is : x2 / a2 - y2/b2 = 1 ( or written in the other way, with y2 as first term ) . This equation , plotted, result in a different type of graphic as I learned on T=constant transformation ! My question is why I used to draw the curbe line graph in p-V coordinates of an equation like y=1/x ? ( as pV= constant ) , saying that it is a hyperbola? What has got to do with an arc of hyperbola? Thank you !

If you have the curve given by ## x^2 - y^2 = 1## and you rotate it 45 degrees around the origin, what do you get?

Iustin Ouatu
As a variant of DEvens's post, suppose you define u=x+y and v=x-y. What is the curve in terms of u and v?

DEvens
Thank you for your responses !
@DEvens :yes, I realized that x^2−y^2=1 shifted by π/4 radians results in my desired part of graph, but... something is not clear in my mind. How can I assume that I can rotate the graph and still get something mathematically valid? There's something not clear in my mind...
@robphy : in terms of u and v, u*v = x^2-y^2 =constant =1 , as in a normal isotherm. But from the (x,y) coordinates , I can define a new system of coordinates, given by (x+y, x-y ) ? Just like when I shift by π/4 radians the normal hyperbola graph?

A hyperbola is shape, not a function.
Focus on the curve, and forget about the axes.

Drawn on a piece of paper, that curve is a hyperbola... no matter how you slide, reflect, or rotate the piece of paper.

Iustin Ouatu

## 1. What are Clapeyron-Mendeleev coordinates?

Clapeyron-Mendeleev coordinates are a coordinate system used in thermodynamics to plot the relationship between temperature and pressure for a substance. They were named after the scientists Benoît Paul Émile Clapeyron and Dmitri Mendeleev.

## 2. What is a hyperbola transformation?

A hyperbola transformation is a mathematical operation that converts a curved line into a straight line. In the context of Clapeyron-Mendeleev coordinates, it allows for the relationship between temperature and pressure to be represented by a straight line, making it easier to analyze and understand.

## 3. How are hyperbola transformations used in thermodynamics?

In thermodynamics, hyperbola transformations are used to simplify the relationship between temperature and pressure for a substance. By transforming the curved relationship into a straight line, it becomes easier to calculate properties such as enthalpy, entropy, and Gibbs free energy.

## 4. What is the significance of hyperbola transformations in Clapeyron-Mendeleev coordinates?

Hyperbola transformations in Clapeyron-Mendeleev coordinates are significant because they allow for a more intuitive and practical representation of the relationship between temperature and pressure for a substance. This makes it easier for scientists to analyze and interpret thermodynamic data.

## 5. Can hyperbola transformations be used for all substances?

Hyperbola transformations can be used for most substances, but they are most accurate for ideal gases. For real substances, the transformation will result in a slightly curved line rather than a perfectly straight line. However, it is still a useful tool for analyzing the relationship between temperature and pressure in real substances.

Replies
1
Views
966
Replies
1
Views
329
Replies
4
Views
412
Replies
2
Views
3K
Replies
1
Views
1K
Replies
2
Views
980
Replies
6
Views
2K
Replies
22
Views
2K
Replies
2
Views
2K
Replies
15
Views
4K