# 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