Question

During mineral formation, the same chemical compound can be come different minerals depending on the temperature and pressure at the time of formation. A phase diagram is a graph that shows the conditions that are needed to form each mineral. The graph above is a portion of the phase diagram for aluminosilicates, with the temperature t , in degrees

Celsius , on the horizontal axis, and the pressure p , in gigapascals (G Pa), on the

Which of the following systems of inequalities best describes the region where sillimanite can form?

Hint:

**Hint:**

We need to find the system of inequalities which describe the region where sillimanite can form. First we find the equation of the boundary line of the region where sillimanite can be formed in the phase diagram. Then we put the inequalities sign in the appropriate equation by observing if the lower or upper part of the line is the region where the sillimanite can be formed.

## The correct answer is:

### The boundaries of the region where sillimanite can be formed are given by two lines: one is the boundary line between kyanite and sillimanite and the other is the boundary between andalusite and sillimanite.

First we find the equation of the boundary line between kyanite and sillimanite:

There are two points lying on this line : and .

Let these points be denoted by

and

In an xy plane, equation of a line passing through two points (a , b ) and (c , d ) is given by

Using the above points, we have

Simplifying the equation, we have

Cross multiplying, we get

We use the calculator to expand the above equation

Taking all the terms in the left hand side, we get

Dividing by 295 throught, we get

As the region which forms sillimanite is below this line, we change the equality to less thanor equal to inequality,

Next,

We find the equation of the boundary line between andalusite and sillimanite:

There are two points lying on this line and

Let these points be denoted by

In an xy plane, equation of a line passing through two points (a , b ) and (c, d ) is given by

Using the above points, we have

Simplifying the equation, we have

Cross multiplying, we get

We use the calculator to expand the above equation

Taking all the terms in the left hand side, we get

Dividing by 260 throught, we get

As the region which forms sillimanite is above this line, we change the equality to greater than or equal to inequality,

Finally, we get

The inequalities which describe the region where sillimanite is formed are

Rewriting these inequalities, we get

Option B)

best describes the region where silliminate can form.

**Note:**

We do not have the exact inequalities in the option, so we choose the one from the options which is the closest approximation of the inequalities that we have calculated. There are a couple of formulas and concepts used here, such as the equation of a line passing through two points and the concept that the region below a line is given by replacing the equality sign with less than or equal to in the standard form of the equation and vice versa.

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During mineral formation, the same chemical compound can beco me different minerals depending on the temperature and pressure at the time of formation. A phase diagram is a graph that shows the conditions that are needed to form each mineral. The graph above is a portion of the phase diagram for aluminosilicates, with the temperature t , in degrees Celsius , on the horizontal axis, and the pressure p , in gigapascals (GPa), on the vertical axis.

An equation of the boundary line between the andalusite and sillimanite regions is approximated by the equation above. What is the meaning of the T-intercept of this line?

**Note;**

A phase diagram represents the different physical states of some substance under different conditions, such as termperature and pressure, graphically. It may seem like a complicated graph but we just needed to focus on the equation of one given line.

During mineral formation, the same chemical compound can beco me different minerals depending on the temperature and pressure at the time of formation. A phase diagram is a graph that shows the conditions that are needed to form each mineral. The graph above is a portion of the phase diagram for aluminosilicates, with the temperature t , in degrees Celsius , on the horizontal axis, and the pressure p , in gigapascals (GPa), on the vertical axis.

An equation of the boundary line between the andalusite and sillimanite regions is approximated by the equation above. What is the meaning of the T-intercept of this line?

**Note;**

A phase diagram represents the different physical states of some substance under different conditions, such as termperature and pressure, graphically. It may seem like a complicated graph but we just needed to focus on the equation of one given line.

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