Animated Solution for Physics - Atoms and Nuclei: For uranium nucleus how does its mass vary with volume ?

Imagine holding a piece of uranium.

At its core lies the nucleus, a dense cluster of protons and neutrons.

The question asks us a very fundamental property about this nucleus: how does its mass relate to its volume?

To understand this, we need to look at the concept of nuclear density.

Density, as we know, is the ratio of mass to volume:

Now, one of the most fascinating discoveries in nuclear physics is that nuclear matter is incredibly dense.

More importantly, this density is practically constant for all nuclei, from the lightest hydrogen to the heaviest uranium.

Since the density $\rho$ is a constant, we can easily rearrange our formula:

Because $\rho$ is just a constant number, we can replace the equals sign with a proportionality sign:

This tells us that as the volume of a nucleus increases, its mass increases by the exact same factor.

Thus, the correct relationship is $m\propto V$, making option (a) the right answer.

You might wonder why this happens.

It comes down to how the size of a nucleus scales with the number of nucleons (protons and neutrons), denoted by $A$.

The radius $R$ of a nucleus is given by:

where $R_0$ is a constant.

If we calculate the volume assuming a spherical nucleus:

So, $V\propto A$.

At the same time, the total mass $m$ is roughly the mass of a single nucleon $m_p$ times the total number of nucleons $A$:

So, $m\propto A$.

Since both mass and volume are directly proportional to the mass number $A$, their ratio (the density) is a constant.

This constant density is mind-bogglingly high—about $2.3\times 10^{17}{\text{ kg/m}}^3$!