Confused (supposedly easy chem question: volume, density)

[uchihajeff]

This doesn't seem like it should be too hard, but I'm stuck... Am i making a stupid mistake somewhere?

1. The density of potassium metal is 0.89 g/cm3. Potassium's atomic radius is 231 pm. Assuming that r nuc = 10^-4 x r atom, estimate the density of an essentially spherical 39K nucleus.

Homework Equations

Volume of a sphere

The Attempt at a Solution

Convert pm to cm: 231 pm = 2.31E-8 cm

Find number of grams in the whole atom by finding the volume of the atom and mulitplying it by the density: (4/3)(pi)(2.31E-8)^3 * .89 = 4.5953E-23 g

Radius of nucleus = 1E-4 * 2.31E-8 cm = 2.31E-12 cm

Volume of nucleus = (4/3)(pi)((2.31E-12)^3) = 5.1632666E-35

Find density by: 4.5953E-23/5.1632666E-35 = 8.899E11 g/cm^3

Unfortunately, that doesn't match with the answer given: The answer given is 1.26E12 g/cm^3
Thanks for any help.

 

[lewdtenant]

there is something wrong.

the "answer" * the volume of nucleus is larger than the total mass of the atom. this can't be.

 

[HallsofIvy]

there is something wrong.

the "answer" * the volume of nucleus is larger than the total mass of the atom. this can't be.

That makes no sense! "Volume" and "mass" are not comparable. It makes no sense to say one is larger than the other!

 

[lewdtenant]

I'm sorry, let me clarify. the supposed answer (which is a density) multiplied by the volume of nucleus is larger than the total mass of the atom.

density times volume gives mass.

 

[Ygggdrasil]

Spheres cannot be packed together w/o leaving empty space, so it's wrong to assume the density of potassium metal is equal to the density of a potassium atom. To get the mass of a ³⁹K atom, just convert 39 amu to grams.