Alloy Lab Report: Solving Equations to Estimate Element Percentages in an Alloy

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Homework Help Overview

The discussion revolves around estimating the percentages of two elements in an alloy using given equations that relate the fractions of each element to the overall density of the alloy. The context involves algebraic manipulation of these equations to derive formulas for the fractions based on measured and known densities.

Discussion Character

  • Exploratory, Mathematical reasoning, Assumption checking

Approaches and Questions Raised

  • Participants discuss the algebraic manipulation of the equations provided, with attempts to express one variable in terms of another. Questions arise regarding the appropriateness of using mass versus density in the calculations, and whether the initial equations are being applied correctly.

Discussion Status

There is ongoing exploration of how to solve the equations with some participants suggesting substitution methods. However, there is no explicit consensus on the best approach, and some participants express uncertainty about the relevance of mass in the context of the problem.

Contextual Notes

Participants note that the problem involves two unknowns and two equations, but there is confusion regarding the definitions of the variables and the proper application of the densities in the calculations. The instructions emphasize using densities without incorporating mass directly.

myelevatorbeat
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First, here is the question:

The percentage of two elements making up an alloy can be estimated from the following equations, which assume a simple mixing of the two components:
Palloy=F1P1+F2P2
F1+F2=1

Here, F1 is the fraction of the alloy composed of element 1, and F2 is the fraction of the alloy composed of element 2. Again, these equations provide only an estimate because the formation of an alloy typically involves processes more complex than simple mixing. Solve these equations algabraically to derive formulas for f1 and f2. Next, plug in your measured density for steel and the spec densities of the two elemental components to estimate f1 and f2, the percentage of each component in the steel alloy of your sphere.

This is what I did so far but I don't really have a clue if it's wrong:

F1=Palloy - F2P2 / P1
F2=Palloy - F1P1 / P2

and F1=1-F2
F2=1-F1

I don't have a clue where to go from here. If someone could give me a hint on how to get started, I would really appreciate it.

So far, I was calculating it by:

x+y = 0.0282 kg (the mass of my steel ball)

x/1700 + y/7874 = 3.5914 x 10^-6

with
1700 =density of carbon
7874 = density of iron
3.5914 x 10^-6 = volume of ball

I don't think this way really follows the above formulas though.

So, would it be better to go 7852=1700x + 7874y
with
7852 = density of my ball bearing
1700=carbon density
7874=iron density

Also, how could I go about solving the above equation since I don't know what x or y are.
 
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Assuming one has used the right equations and values,

x+y = 0.0282 kg --> x = 0.0282 - y or y = 0.0282 - x
7852=1700x + 7874y

One has two equations and two unknowns.
 
Yeah, but somehow I'm supposed to solve those equations and get the answers to both variables and I just can't see how I'm going to do that.
 
Using either x = 0.0282 - y or y = 0.0282 - x

Use substitution into 7852=1700x + 7874y, which will give one an expression in only one variable. Then knowing the one value for x or y, substitute that value into either equation and solve for the other value.
 
Why would I use the mass though? The instructions tell me I should only use the 3 densities and be able to solve it.
 
Be careful about changing the definition of x and y.

If one uses just density, then for 7852=1700x + 7874y, x and y are fractions (on a mass basis), and y = 1-x. One can put density on a mass basis by multiplying by unit volume.