Question

The ratio of atomic volume of nuclear volume is of the order of (A) \(10^{-15}\) (B) \(10^{-10}\) (C) \(10^{15}\) (D) \(10^{-10}\)

Short answer

The ratio of atomic volume to nuclear volume is of the order of \(10^{15}\).

Step-by-step solution

1. Find the atomic volume

To find the atomic volume, remember that the majority of the volume of an atom is mostly empty space, with the electrons orbiting the nucleus at a distance. The radius of an atom (called the atomic radius) is typically around 1 Ångström (1 Å = \(10^{-10}\) meters). The volume of a sphere can be calculated using the formula: \[V = \frac{4}{3}\pi r_{a}^3\] Here, the atomic radius is denoted by \(r_{a}\) and is approximately \(10^{-10}\) meters.

2. Find the nuclear volume

To find the nuclear volume, understand that the nucleus is much smaller than the atom itself. The nuclear radius (denoted by \(r_{n}\)) is approximately \(10^{-15}\) meters. Again, we can use the formula for the volume of a sphere: \[V = \frac{4}{3}\pi r_{n}^3\]

3. Calculate the ratio of atomic volume to nuclear volume

Now we'll calculate the ratio of the atomic volume to the nuclear volume. \[\frac{V_{a}}{V_{n}} = \frac{\frac{4}{3}\pi r_{a}^3}{\frac{4}{3}\pi r_{n}^3}\] Since both atomic and nuclear volumes are spheres, we can simplify the equation by cancelling out the \(\frac{4}{3}\pi\) terms: \[\frac{V_{a}}{V_{n}} = \frac{r_{a}^3}{r_{n}^3}\] Now, substitute the values of the atomic and nuclear radii: \[\frac{V_{a}}{V_{n}} = \frac{(10^{-10})^3}{(10^{-15})^3}\] This ratio simplifies to: \[\frac{V_{a}}{V_{n}} = \frac{10^{-30}}{10^{-45}}\] After dividing the exponential terms, we are left with: \[\frac{V_{a}}{V_{n}} = 10^{15}\] So, the ratio of the atomic volume to the nuclear volume is of the order of \(10^{15}\), which corresponds to option (C).