# The density of a proton (hydrogen nucleus)

• mrcake
In summary, the conversation discusses calculating the volume and density of a proton using the formula V = 4/3 x pi x r^3 and D = m/V. The final result is a density of 2.3 x 10^17 kg/m^3, which is questioned for its meaning and relevance. f

#### mrcake

Homework Statement
A proton, which is the nucleus of a hydrogen atom, can be modeled as a sphere with a diameter of 2.4 fm and a mass of 1.67 x 10^-27 kg. Determine the density of the proton
Relevant Equations
D = m/V
First of all i want is to check if my answer is right or not because i am really not sure about my answer.
Because the length is given in the form of a diameter we will divide that by 2 so we get it in the form of radius,
2.4fm / 2 = 1.2fm = r
Then we will convert the radius from "fm" to "m" so we that'll be,
r = 1.2 x 10^-15 m.
so we now have the radius in meters and from that we can calculate the volume of the proton, by the following formula V = 4/3 x pi x r^3
V = 4/3 x pi x (1.2 x 10^-15)^3 = 7.23 x 10^-45 m^3
Since we have the mass and we calculated the volume we can plug them into formula of density so we'll get,
D = m/V
D = 1.67 x 10^-27 / 7.23 x 10^-45 = 2.3 x 10^17 kg/m^3

Something is not right with your answer. Take a good look at it. Hint: Water has density 1000 kg/m3.

hi thanks for reply, but what does water have to do with the proton of the hydrogen atom

PeroK
How many protons does a glass of water (0.25 L) contain? What would be the mass of that glass of water if your number for the density is correct?

How many protons does a glass of water (0.25 L) contain? What would be the mass of that glass of water if your number for the density is correct?
Water is mostly space. A single proton may have a huge classical density - compare with a neutron star, for example.

D = 1.67 x 10^-27 / 7.23 x 10^-45 = 2.3 x 10^17 kg/m^3
Looks about right given the numbers.

mrcake
Looks about right given the numbers.
Is it right?

Is it right?
Google thinks it's right to within an order of magnitude.

Is it right?
It seems a fairly meaningless number, if you ask me. The density of water or hydrogen gas is meaningful; but the density of an elementary particle, given that it isn't actually a solid, localised sphere seems a pointless calculation. Sorry for the philosophical answer.

berkeman