Question

Calculate number of neutrons present in \(12 \times 10^{25}\) atoms of oxygen \(\left({ }_{8} \mathrm{O}^{17}\right):\) (Given: \(\left.N_{A}=6 \times 10^{23}\right)\) (a) 1800 (b) 1600 (c) \(1800 \mathrm{~N}_{\mathrm{A}}\) (d) \(3200 \mathrm{~N}_{A}\)

Short answer

Total number of neutrons present in 12 \times 10^{25} atoms of oxygen-17 is \(1800 N_{A}\).

Step-by-step solution

Identify the number of neutrons in one atom of Oxygen-17

An atom of oxygen-17 is represented by \( {}_{8}O^{17} \), where 8 is the atomic number (number of protons) and 17 is the mass number (total number of protons and neutrons). To find the number of neutrons, subtract the atomic number from the mass number: \[\text{Number of neutrons} = \text{Mass number} - \text{Atomic number} = 17 - 8 = 9.\]

Determine the total number of neutrons in all atoms

To get the total number of neutrons in all atoms, multiply the number of neutrons per atom by the total number of atoms: \[\text{Total neutrons} = \text{Number of neutrons per atom} \times \text{Total number of atoms} = 9 \times 12 \times 10^{25}.\]

Simplify the expression

Simplify the multiplication to find the total number of neutrons: \[9 \times 12 = 108,\] and then \[108 \times 10^{25} = 108 \times 10^{2} \times 10^{23} = 10800 \times 10^{23}.\]

Relate to Avogadro's number

Given that Avogadro's number \(N_{A} = 6 \times 10^{23}\), express the total number of neutrons in terms of \(N_{A}\): \[\text{Total neutrons} = \frac{10800 \times 10^{23}}{6 \times 10^{23}} \times N_{A} = 1800 \times N_{A}.\] Hence, the total number of neutrons can be represented as \(1800 N_{A}\).

Key concepts

Avogadro's Number

Avogadro's number, often denoted as \(N_A\), is a fundamental constant in chemistry and physics that represents the number of particles, typically atoms or molecules, in one mole of a substance. A mole is one of the base units in the International System of Units (SI) and it is defined as the amount of a substance that contains as many particles as there are atoms in exactly 12 grams of carbon-12.

The value of Avogadro's number is approximately \(6.022 \times 10^{23}\). It serves as a bridge between the microscopic scale, where atoms and molecules exist, and the macroscopic scale, which we can measure in the laboratory. For example, when you have a mole of oxygen atoms, you have \(6.022 \times 10^{23}\) oxygen atoms.

Understanding Avogadro's number is crucial when converting between number of atoms and moles, as it enables chemists to quantify atoms - which are incredibly small - in large enough amounts to be useful for chemical reactions and stoichiometry calculations.

Mass Number

The mass number, symbolized as \(A\), of an atom represents the total number of protons and neutrons in its nucleus. Since protons and neutrons are much heavier than electrons and contribute almost all of an atom's mass, the mass number approximately equals the atomic mass. For instance, in an atom of oxygen-17, represented as \({ }_{8}O^{17}\), the '17' is the mass number, indicating 17 protons and neutrons combined.

However, it's important to clarify that the mass number is not the same as atomic mass, which is the averaged mass of all the naturally occurring isotopes of an element, measured in atomic mass units (amu). The mass number is always a whole number whereas the atomic mass, due to isotopic mixtures and the binding energy of the nucleus, may not be.

Determining the mass number is essential when calculating the number of neutrons in an atom by subtracting the atomic number (the number of protons) from the mass number.

Atomic Number

The atomic number, represented by \(Z\), is a fundamental property of an element and its atoms that denotes the number of protons in the nucleus of an atom. On the periodic table, elements are arranged in order of increasing atomic number. For example, oxygen has an atomic number of 8, which means every oxygen atom has 8 protons.

The atomic number is significant not only because it defines the element, but also because it concludes the chemical behavior of the atom. This is due to the fact that the number of protons determines the electronic structure of an atom, which dictates how it will interact with other atoms.

Furthermore, the difference between the atomic number and the mass number gives us the number of neutrons in a neutral atom, since the overall charge must balance out with electrons. Thus, the atomic number is both an identifier for the element and a crucial starting point for understanding the structure of atoms.

Isotopes of Oxygen

Isotopes are variants of a particular chemical element that have the same number of protons (and hence the same atomic number) but different numbers of neutrons, resulting in different mass numbers. Although isotopes of an element exhibit nearly identical chemical properties because they have the same electron configuration, their physical properties may be different due to their varied mass.

Oxygen has several isotopes, with the most abundant being oxygen-16 (\(^{16}O\)), which has 8 protons and 8 neutrons. Other isotopes include oxygen-17 (\(^{17}O\)) and oxygen-18 (\(^{18}O\)), with 9 and 10 neutrons respectively. These isotopes are important in various scientific fields, including climatology, where ratios between oxygen-18 and oxygen-16 in ice cores are used to deduce past temperatures on Earth.

In exercises that involve oxygen isotopes, understanding that isotopes have a common atomic number but different mass numbers is key to performing accurate calculations, such as finding the number of neutrons by subtracting the atomic number from an isotope's mass number.