The Wire Problem & Dimensional Analysis

Seeing as how I’m not a professor, I don’t have to follow the same rules as your chemistry professor so this is basically a step by step solution to this problem. I am warning you against just using this solution to enter the correct answer in to your homework, because come test time you won’t remember it. Try instead to break it down in your head, picture every step and make sure you understand it before moving on to the next step.

A prime example of the type of problem that you will be dealing with in chemistry is as follows (more or less).

Copper can be drawn into thin wires. How many meters of 34-gauge wire (diameter = 6.304 x 10-3 in) can be produced from copper in 5.01 lbs of covellite, an ore of copper that is 66% Cu by mass? Copper has a density of 8.95 g/cm3. Assume that the wire is a cylinder, the formula for Volume of a cylinder is V=π r2h

This is a dimensional analysis problem, in essence you are given different dimensions and you are trying to figure out which one you are missing to solve the problem.

Your first task in all of this is going to be to figure out what information is relevant and how can you start to set up the problem. My suggestion would be to start with the fact that the problem is TELLING YOU THAT THE WIRE IS A CYLINDER! Sorry, didn’t mean to shout, just trying to make a point. Once you know that you’re dealing with a cylinder then you can start to gather what you know about that cylinder and eliminate variables.

*suggestion – draw it out

drawing cylinder.png

Once you have your picture drawn, it’s time to start figuring out what information you have about that cylinder. For example, there are two bits of information that you already have. You’ll need to convert them and play with them a little bit, but they’re there. Anyone? No?

Think about it; what is density (I don’t expect you to have that memorized yet, but you will)? Density=mass/volume. What else? What is the radius? How much copper do you actually have? Before moving on WRITE IT OUT.

what is given.png

Hint: When problem solving in chemistry, always, always, always convert to SI units. The conversions for these units have to be given, or it has to be implicitly stated during the course that they should be memorized.

Okay, so now that I have all information written out, I’m going to figure out what conversions I’m going to need to make, remember you need SI units.

Since you are given the diameter of the cylinder you now need to know if it’s in the correct unit. Last time I checked inches isn’t an SI unit so I’m going to need to convert. In this case we are given an entire ratio in SI units. The “Volume” in density is given in cm3 so I’m going to assume that I’m going to have to convert inches to centimeters. Through the power of Google I now know that:

1 inch = 2.54 centimeters

Now for the fun part, conversion.

converting inches to cm in diameter.png

You can put this in scientific notation, but be careful about your significant figures, I always opt for using more sig-figs during the problem solving and then bringing them back down at the very end.

Ask yourself, why the centimeters is on top and why the inches is on the bottom? I like nice whole numbers when I do my calculations so I will always opt for using them when I can. You can actually use the fact that 1cm=0.393701in, but that’s a pain in the ass and it requires a different set up of the problem.

Now that we have diameter in centimeters, let’s put it aside for now and move on to a more complex part of the problem. You are trying to make a pure copper wire, but covellite isn’t pure, it’s only 66% copper by mass. You’re going to need to isolate the amount of copper from the covellite, but you’re also going to need to convert it over to SI units. I can’t really explain it in words, so we just have to work it out.

*Suggestion-go to SI units first then figure out your mass of copper. 1lb = 453.592g

covellite conversion to SI .png

Now that we’ve converted over to SI units we can calculate how much copper we have in the covellite. When we’re using percentages for calculations we always work them based on a unit of one; where 1 represents 100% of the thing being analyzed. So 8% becomes .08, 66% becomes .66, so on and so forth.

extracting copper.png

At this point it’s important to ask why the unit of “grams” didn’t cancel out. I’m going to leave that up to you to figure out, but just know that they don’t.

You are almost there guys, just hang in there. You’re next step is going to be to look at the equation for the volume of a cylinder again. You have now successfully isolated the amount of copper that you have and you have converted everything over to SI units. So where does density come in to play? Remember volume? Yup, you’re given that variable and since you know your diameter and therefore your radius the only variable that’s left is the height! That’s what you’re trying to solve for! The question now becomes how do we get volume from the mass of copper?

If you know what the “d” in the density formula is and you have just figured out the mass of the copper, just manipulate the density equation to isolate “V” and plug and play!

figuring out volume.png

*Correction- it is the volume of the mass of copper that you’re working with, not the volume of the cylinder just yet.

Here you are, you’ve almost made it, there’s light at the end of the tunnel. Now all you have left to do is manipulate the original equation for the volume of a cylinder to isolate the variable that you want and plug-in your numbers and you’ll be done!

*Having said that, make sure you use radius not diameter in your equation, and make sure all of your units cancel appropriately.

Length of your wire.png

BUT WAIT!!! THERE’S MORE!!!!!

After all of that it’ll be easy to forget that you’re being asked to give your answer in meters. I think you can all figure that out.

I hope this helps you guys understand dimensional analysis a little bit more. Remember you’re dealing with dimensions! Ask yourself what this means and look up videos on it to help you better understand it. If you still have questions feel free to email me, or if you’re a Concordia student, visit me at my “TA Hours“. Happy Sciencing!

-R

[Chemistry 211 Chapter 1 – The Wire Problem]

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s