Question

The radius of an atomic nucleus is of the order of: (a) \(10^{-8} \mathrm{~cm}\) (b) \(10^{-13} \mathrm{~cm}\) (c) \(10^{-10} \mathrm{~cm}\) (d) \(10^{-15} \mathrm{~cm}\)

Short answer

The radius of an atomic nucleus is of the order of \(10^{-13} \, ext{cm}\), so the correct answer is (b).

Step-by-step solution

Understanding the Problem

We need to identify the order of magnitude that correctly represents the radius of an atomic nucleus. We're given four options, each a different power of ten in centimeters.

Recalling Atomic Nucleus Size

Atomic nuclei are typically very small compared to atoms, and their size is generally around a few femtometers. A femtometer is equal to \(10^{-15}\) meters.

Converting Units

Since the problem uses centimeters, we need to convert femtometers to centimeters. Knowing that 1 meter is 100 centimeters, \(1 \, ext{femtometer} = 10^{-15} \, ext{meters} = 10^{-15} imes 100 \, ext{cm} = 10^{-13} \, ext{cm}\).

Selecting the Closest Option

From the calculations, the radius of an atomic nucleus is on the order of \(10^{-13}\) cm. Therefore, the correct choice among the options is (b) \(10^{-13} \, ext{cm}\).

Key concepts

Atomic Nucleus

An atomic nucleus is the central core of an atom. It contains most of the atom's mass and is composed of protons and neutrons, collectively known as nucleons. Unlike the expansive electron cloud surrounding it, the nucleus is incredibly tiny. Despite its small size, it holds almost all the atom's mass due to the relative heaviness of protons and neutrons.

Atomic nuclei are measured in femtometers because they are so diminutive. Understanding the size and composition of atomic nuclei is essential for grasping atomic structure and nuclear physics. Nuclei are responsible for the atomic number, not only defining the type of element but also influencing its chemical behavior.

This understanding is fundamental in fields ranging from basic chemistry to advanced physics, and even medicine, such as in the use of isotopes for diagnostic imaging.

Order of Magnitude

Understanding the order of magnitude is crucial in physics and science in general. It describes the scale of a quantity relative to a factor of ten. This concept allows scientists to simplify complex numerical problems by focusing on the scale instead of precise numbers.

When dealing with atomic structures like the nucleus, it's common to use orders of magnitude to speak about their size. In this context, the radius of an atomic nucleus is typically around a few femtometers, which corresponds to an order of magnitude of \(10^{-15}\) meters. In the original exercise, this is further converted to centimeters, giving us an order of magnitude of \(10^{-13}\) cm. This simplification helps identify the closest approximation among possible answers without getting bogged down by complex numbers.

Unit Conversion

Unit conversion is a key skill in scientific calculations. It allows one to move measurements from one unit to another, ensuring precision and consistency across disciplines, like physics and chemistry.

Converting units may involve simple multiplication or division. For example, in the exercise, we started with units in femtometers and needed to convert them to centimeters. Knowing that 1 meter equals 100 centimeters, we transform 1 femtometer (\(10^{-15}\) meters) to \(10^{-13}\) cm. This involves multiplying the femtometer measurement by the meter-to-centimeter conversion factor.

Such conversions facilitate our understanding of physical quantities in units familiar to us, making complex scientific concepts more approachable.

Femtometer

A femtometer is a unit of length in the metric system, specifically representing \(10^{-15}\) meters. It is primarily used when measuring atomic and subatomic structures, such as atomic nuclei.

The size of a femtometer helps scientists describe and understand the minute world of nuclear physics. Paper-thin in measurement, it is succinctly useful when discussing things that are inconceivably tiny. For example, a proton's radius is about 0.84 femtometers.

Given the scale at which femtometers operate, they highlight just how small and compact the atomic nucleus is compared to the entire atom. This unit of measurement not only supports precision in scientific dialogues but also helps bridge the vast gap between the everyday macroscopic world and the microscopic scale at which nuclear physics operates.