# Gold Atom Count in O-Ring: 10.4x10^21

• shreddinglicks

## Homework Statement

A golf O-ring is used to form a gastight seal in a high-vacuum chamber. The ring is formed from a 100-mm length of 1.5-mm-diameter wire. Calculate the number of gold atoms in the O-ring.

## Homework Equations

Volume of cylinder = PI*h*r^2

## The Attempt at a Solution

I got the volume of the cylinder:
PI*100*.75^2 = 176.7

I used the formula m = D*V:

density gold = 19.28 g/cm^3
amu gold = 196.97 grams

19.28*176.7 = 3406.776

I then multiplied by avogadro's number over amu.

3406.776 *(6.02x10^23 / 196.97) = 1.04x10^25 atoms

The answer should be 10.4x10^21 atoms. I'm thinking I messed up some unit conversion. I am unsure of where.

## Homework Statement

A golf O-ring is used to form a gastight seal in a high-vacuum chamber. The ring is formed from a 100-mm length of 1.5-mm-diameter wire. Calculate the number of gold atoms in the O-ring.

## Homework Equations

Volume of cylinder = PI*h*r^2

## The Attempt at a Solution

I got the volume of the cylinder:
PI*100*.75^2 = 176.7

I used the formula m = D*V:

density gold = 19.28 g/cm^3
amu gold = 196.97 grams

19.28*176.7 = 3406.776

I then multiplied by avogadro's number over amu.

3406.776 *(6.02x10^23 / 196.97) = 1.04x10^25 atoms

The answer should be 10.4x10^21 atoms. I'm thinking I messed up some unit conversion. I am unsure of where.

What are the units on your volume? What are the units on your density?

The density is g/cm^3

the volume is mm^2

I see the issue, how do I convert this? Is there a way to convert so I can cancel out a unit?

The density is g/cm^3

the volume is mm^2

I see the issue, how do I convert this? Is there a way to convert so I can cancel out a unit?

Volume should be cubic something. mm^2 is an area.

To convert a volume in mm^3 to cm^3 think about the following:

Conversion factor * X mm^3 = Y cm^3

The conversion factor needs to have units of cm^3 on the top and mm^3 on the bottom. What about [( 1 cm/ 10 mm)]^3 ? I.e. can you use the conversion between cm and mm to come up with a conversion factor between mm^3 and cm^3 ?

I see, so I have [176.7 mm^3] * [1cm/10mm]^3

basically

176.7 / 10^3 = .1767

Numbers are OK, but in the final formula you again ignored units. That's the simplest way of making mistakes, as you have already seen. Don't do that.

.1767cm^3

Now use this number to answer the problem.