Solve Gold Density Problem: Area & Length of Gold Leaf & Fiber

[JudyyNunez]

1. Problem: Gold, which has a density of 19.32 g/cm3, is the most ductile metal and can be pressed into a thin leaf or drawn out into a long fiber. (a) If a sample of gold with a mass of 2.274 g, is pressed into a leaf of 8.678 μm thickness, what is the area (in m2) of the leaf? (b) If, instead, the gold is drawn out into a cylindrical fiber of radius 2.200 μm, what is the length (in m) of the fiber?


2. Homework Equations : a=pi x r^2 and (a)(l)= part a



3. The Attempt at a Solution : part a= 0.01356, and I tried plugging it into the equation (1.521x10^-11 m^2)(l)=0.01356

If anyone could find the error in part B! Please Help

 

[SteamKing]

Units? Showing your work?

 

[AlephZero]

2. Homework Equations : a=pi x r^2 and (a)(l)= part a

The "(a)(l)= part a" looks wrong. The answer to part a is an area. You can't say "area times length = area".

3. The Attempt at a Solution : part a= 0.01356

That looks right (except you forgot the units).

I don't understand what you did for part (b).

 

[NascentOxygen]

what is the area (in m2) of the leaf?

Don't forget that the leaf has two sides, so there are two surface areas. This could be a trick question!

So state your answer as ... m² per side for each of 2 sides.

If you knew the shape, you could also work out the area of its other very narrow side/s.

 

[HalfLostAndSearching]

For part a, the correct equation to use is A = m/d, where A is the area, m is the mass, and d is the density. So, the area of the leaf is 2.274 g / (19.32 g/cm3 x 8.678 μm) = 1.521 x 10^-11 m^2.

For part b, the correct equation is V = πr^2l, where V is the volume, r is the radius, and l is the length. We can rearrange this equation to solve for l: l = V / (πr^2). The volume of the cylindrical fiber can be found using the density equation, d = m/V, where d is the density, m is the mass, and V is the volume. So, the volume of the fiber is 2.274 g / 19.32 g/cm3 = 0.1175 cm3. Converting this to m^3, we get 1.175 x 10^-7 m^3. Plugging this into the equation for l, we get l = (1.175 x 10^-7 m^3) / (π x (2.200 μm)^2) = 1.035 m.