Question

Determine the approximate ratio of the diameter of a uranium nucleus \((A=238)\) to that of a beryllium nucleus \(\\{A=9\rangle\)

Short answer

The ratio of the diameters of the uranium nucleus to the beryllium nucleus given the atomic mass numbers 238(A for Uranium) and 9(A for Beryllium) can be determined by finding the cube root (\(A^{1/3}\)) of each atomic mass number. Therefore, the ratio of diameters \(Uranium : Beryllium\) is the cube root of the ratio \(238 : 9\)

Step-by-step solution

Apply the Diameter Formula for Uranium Atom

Compute the diameter of the uranium atom using the formula \(D = D_{0} A^{1/3}\). As \(D_{0}\) is a constant, for comparison purpose, we can let \(D_{0} = 1\). The Uranium atom diameter \(D_{u}\) will then be \(D_{u} = A_{u}^{1/3}\), where \(A_{u}\) is the atomic mass number of Uranium, which is 238.

Apply the Diameter Formula for Beryllium Atom

Similarly, compute the diameter of the Beryllium atom using the formula \(D = D_{0} A^{1/3}\). The Beryllium atom diameter \(D_{b}\) will then be \(D_{b} = A_{b}^{1/3}\), where \(A_{b}\) is the atomic mass number of Beryllium, which is 9.

Compute the Ratio

Next, find the ratio of the diameters of Uranium atom to Beryllium atom. \(Ratio = \frac {D_{u}}{D_{b}}\)

Key concepts

Atomic Mass Number

The atomic mass number, symbolized as 'A,' is crucial in understanding the structure of an atom's nucleus. It represents the total number of protons and neutrons within the nucleus. Protons carry a positive charge, while neutrons are neutral, and together they account for almost all of an atom's mass. The atomic mass number is significant because it not only indicates the mass of the atom but also its stability and how it interacts with other atoms.
For instance, in our two subjects of interest, uranium has an atomic mass number of 238, and beryllium has a significantly smaller number of 9. These figures indicate the difference in mass and the number of components within their respective nuclei. A larger atomic mass number typically means a larger nucleus, thus impacting the diameter of the nucleus and the atom as a whole.

Uranium Nucleus

The uranium nucleus is known for being one of the heaviest naturally occurring elements, with an atomic mass number of 238. Uranium is widely known for its role in nuclear reactors and weaponry due to its radioactive properties and ability to sustain a nuclear chain reaction.

Composition of the Uranium Nucleus

Consisting of 92 protons and 146 neutrons, the uranium nucleus is not just massive but also quite complex. Its large atomic mass has direct implications on its size, which becomes important when comparing it with smaller nuclei such as that of beryllium. Understanding the size of the uranium nucleus is essential in fields such as nuclear physics and energy production.

Beryllium Nucleus

Counterpart to the heavy uranium, the beryllium nucleus has a modest atomic mass number of 9. Beryllium, found fourth in the periodic table, has just 4 protons and typically 5 neutrons.

Characteristics of Beryllium

Despite having a smaller atomic mass number, beryllium has significant applications in industries such as aerospace, for its high strength-to-weight ratio, and in nuclear reactors, as a neutron reflector or moderator due to its low atomic number. The size of the beryllium nucleus is much smaller than that of uranium, which expresses the differences that atomic mass numbers can yield in physical properties of the elements.

Nucleus Diameter Formula

The nucleus diameter can be approximated using a simple empirical formula. The formula typically used is: \[D = D_0A^{1/3}\],where 'D' represents the diameter of the nucleus, 'D_0' is a constant approximately equal to 1.23 x 10^-15 meters (which in some problems may be considered unity for simplicity), and 'A' is the atomic mass number. This cubed root relation implies that as atomic mass numbers increase, so does the diameter, but at a decreasing rate.

Applying the Formula

When comparing the uranium nucleus to the beryllium nucleus using this formula, it is apparent that the considerably larger atomic mass number of uranium results in a considerably larger diameter. However, this increase is not linear due to the cubed root; it illustrates how additional protons and neutrons contribute less to the increase in size as the nucleus becomes larger. This knowledge is valuable when examining the properties of various atoms or in the designing of nuclear materials.