Determining Constants in Equataions

[BayernBlues]

Homework Statement

In all the following problems state the variables or combination of variables
which should be plotted to check the suggested variation and state how the unknown
may be found ( through the slope and/or intercept of the best fitting straight line
y = m x + b.). Because you are not given any numerical data it is just required to
qualitatively describe you method of solution in few lines with schematic graphs.

here are a couple of examples:
The gas law for an ideal gas is PV = RT , P and T are measured variables, V is fixed and
known. Determine R.

The linear expansion of a solid is described by ℓ = ℓo ( 1 + α.Δt ) where ℓ and Δt are
measured variables, ℓo is constant but unknown. Determine α .

The Attempt at a Solution

For the first one:
An increase in T will yield an increase in P because V is a fixed value. P is therefore the dependent variable while T is the independent variable. R is the coefficient to T therefore it will be the slope when P vs T is graphed.

My question is mainly how would I go about problems like this or more complicated than this. When it gives you an equation, two measured values, and a third one which is supposed to be a constant like g. How would you determine the constant?

 

[anathema]

For the second one:
As temperature increases, the length of the solid will also increase due to thermal expansion. In this case, Δt is the independent variable while ℓ is the dependent variable. The coefficient α represents the rate of expansion and can be determined by plotting ℓ vs Δt and finding the slope of the best fitting straight line. To determine ℓo, we can use the intercept of the line, which will be equal to ℓo(1+αΔt=0), therefore ℓo can be found by rearranging the equation to ℓo=-αΔt and substituting a known value for Δt.

 

[m2003]

To determine the constant in an equation, you will need to manipulate the equation to isolate the constant on one side. In the first example, the gas law equation can be rearranged to solve for R by dividing both sides by VT, giving R = P/VT. This shows that R is equal to the slope of a graph of P vs T, since P is the dependent variable and VT is the independent variable. Similarly, in the second example, the equation for linear expansion can be rearranged to solve for α by dividing both sides by ℓoΔt, giving α = (ℓ-ℓo)/ℓoΔt. This shows that α is equal to the slope of a graph of ℓ vs Δt, since ℓ is the dependent variable and Δt is the independent variable. In more complicated equations, you may need to use algebraic manipulation or substitution to isolate the constant and determine its value from a graph.