There are a number of useful relationships in science between changing entities called variables which are described as linear where an incremental change in one variable is matched by an equal incremental change in the other. If one variable doubles in value the other variable will likewise double in value. If one variable decreases to half its size so does the other variable. The ratio between the incremental change in one variable call it the Y variable and the incremental change in another variable related to it call it the X variable is defined as the slope of the curve. If the incremental change is always the same as it will be in a linear relationship then the slope will be a constant value.

In terms of calculus, the incremental change in the Y variable in respect to the incremental change in the X variable will be the difference between two y values on the curve divided by the difference between two cooresponding X values or

dy/dx = Y₂-Y₁ / X₂-X₁

Derivation of the Slope Intercept Formula

Suppose we had a linear curve segment that was placed within a two dimentional Cartesian Coordinate System (refer to Figure 1 below). Lets suppose that this line segment crossed the Y axis at point B where the (x,y) coordinate was (0,B). The other end of the line segment was at coordinates (x,y). If we project the two ends of the line segment so the projections are perpindicular to one another so as to form a right triangle, the perpindicular intersection would be at (x,y) coordinates (x,B). Previously, we showed that the slope was the ratio between the incremental change in Y and the incremental change in X or:

slope = Y₂-Y₁ / X₂-X₁

Since we are dealing with a right triangle then the Trig function equal to that ratio would be equal to the slope. That trig function is the tangent so:

slope = tan alpha = side opposite alpha / side adjacent to alpha.

The side opposite alpha is Y₂-Y₁ and the side adjacent is X₂-X₁

Alpha is the angle formed by the horizontal base of the right triangle and the curve segment itself called the hypotenuse of the right triangle.

Now lets apply this to our specific line segment. The side opposite would be Y-B and the side adjacent would be X-0 so:

slope = Y-B / X-0. Let's call the slope the letter m so:

m = Y-B / X-0 If we solve this for Y:

Y = mX + B

"B" remember is what we call the "Y intercept" that is the y value that the line will have when X = 0 at that same point. In other words where the line crosses or intercepts the Y axis. "m" is the slope which is equal to Y₂-Y₁ / X₂-X₁.

           Y axis (ordinate)
           |        / | (x,y)
           |      /   |
           |    /     |                     slope = m= tan base angle = Y-B / X-0
           |  /       |                               m(X-0) = Y-B
(0,B)   |/____ |  (X,B)                     mX = y-B
           |                                           mX + B = Y
           |
           |
           |__________________________ x axis (abscissa)

                            Figure 1

This derivation above is referred to as the slope-Intercept formula, and it is used extensively in deriving algebraic expressions from linear relationships.

The Slope-Intercept Formula in Chemistry

In Chemistry, we have the linear relationship between the mass of a pure substance and its volume. Suppose that we make an investigation between the mass volue relationship by taking five differently shaped objects all made of the same pure substance. Let's further suppose that we determined the mass and corresponding volume of each object and reported our findings in tabular form. Furthermore, suppose that we plotted the mass of each object on the Y axis with the volume on the X axis and formed five coordinate plot points. We would find that these plot points tended to line up so that a straight line extension would pass through each of them. The line segment formed would intercept the Y axis at coordinates (0,0). If we now applied the slope-intercept formula:

Y = mX + B where B = 0 and m = slope = Density then

mass = Density(Volume) + 0 or

mass / volume = Density

Suggested Exercise

Assuming that the relationship between Celsius temperature readings and Fahrenheit readings is linear withthe linear curve interceting the Y axis at (0,32). Another point on that curve is found to be (100,212).

From this data determine:

Slope of the linear relationship

Derive the algebraic expression using the straight line formula that was previously derived.

Other linear relationships in Chemistry are:

• Relationship between Vapor Pressure of a substance and the temperature
• Relationship between the Equilibrium Constant and temperature
• Relationship between the Kinetic rate constant and temperature
• Relationship between Molar concentration of reactant and time
• Relationship between the Log of Molar concentration and time
• Relationship between the inverse of Molar concentration and time.

These are only a few of the linear relationships in which their algebraic expressions can be derived using the slope-intercept formula.

Linear Relationships in Physics

Physics also have linear relationships such as the relationship between distance of displacement and time of displacement in which the velocity is constant.

R. H. Logan, Instructor of Chemistry, Dallas County Community College District, El Centro College.

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Acknowledgements

Send Comments to R.H. Logan: rhlogan@ix.netcom.com

All contents copyrighted (c) 1995
R.H. Logan, Instructor of Chemistry,DCCCD
All Rights reserved

Revised:11/9/97

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