An atom consists of a tighly bounded nucleus sourrounded by a cloud of electrons. This solar-system model was discovered by Rutherford whom was studying the scattering of alpha particles and was an improvement from the soup model suggested by Thompson where protons and electrons, in equal quantities, were uniformly distributed. The solar-system model was further refined by Bohr specifying that electrons exist in discrete energy orbits (later refined as electron clouds where the cloud represents the probability density of the electron). The strong nuclear foce keeps the positively charged nucleus together.
The nucleus is made up of protons and neutrons. The total number of protons is known as the atomic number, \(Z\), and determines the identity of the atom. The charge of the nucleus is \(+Ze\) since each proton has a charge \(e\) and there are \( Z\) of them. Electrons also contribute to the charge of an atom. In a neutron atom, there are an equal number of electrons and protons and the atom has no charge (i.e. \( -Ze\)).
The number of neutrons is \(N\), the neutron number. The atomic mass, or nucleon number, is the sum of the number of protons, the atomic number, and the number of neutrons, the neutron number, and is given by \(A = Z + N\). Recall that protons and nuetrons have roughly the same mass whereas electrons are much ligher.
Various atoms with different numbers of protons and nuetrons are nuclides. A nuclide is denoted by the chemical symbol (such as \(\text{H}\), \(\text{C}\), \(\text{Sm}\), etc.) with the atomic mass, \(A\) as a superscript such as \(\ce{^4He}\). For further clarity, the atomic number, \(Z\) , is sometimes written as a subscript such as \(\ce{^4_2He}\).
Remember, the atomic number,\(Z\), is unique to a certain atom. However, the number of neutrons may differ. Atoms with the same atomic number but different atom mass are called isotopes (for example, \(\ce{^16_8O}\) and \(\ce{^18_8O}\)). Atoms with the same atomic number,\(Z\), and atomic mass, \(A\), but different mass/energy are isomers (i.e. in an excited state). Unstable isotopes are radioactive and decay into stable atoms using different decaying methods. Isotopes occur naturally in different abundancies given in atomic percentage (or \(\text{a/o}\)).
The atomic weight of an atom is defined as the mass relative to \(\ce{^12C}\) which is taken arbitrarily as \(12\). Formally,
\(M \! \left(\ce{^AZ}\right) = 12 \cdot \cfrac{m \! \left(\ce{^AZ}\right)}{m \!\left(\ce{^12C}\right)}\)
Isotopes that are naturally occuring have an atomic weight equal to the weighted average of the mixture of all isotopes:
\(M = \cfrac{\gamma_i}{100} \sum M_i\)
Given the data in the table below, calculate the atomic weight of naturally occuring oxygen.
| Isotope | Abundance | Atomic Weight |
|---|---|---|
| \(\ce{^16O}\) | \(99.759\) | \(15.99492\) |
| \(\ce{^17O}\) | \(0.037\) | \(16.99913\) |
| \(\ce{^18O}\) | \(0.204\) | \(17.99916\) |
We can use the following equation to determine the atomic weight of the isotope mixture:
\(M = \cfrac{\gamma_i}{100}\sum M_i\)
\(M = \cfrac{1}{100}((15.99492)(99.759) + (16.99913)(0.037) + (17.99916)(0.204)) \)
\(M = 15.9938 \; [\text{amu}]\)
Similarly, the molecular weight is the ratio relative to \(\ce{^12C}\) and is the sum of atomic weights of the atoms that make up the molecule.
\(M(H_2O) = 2 \cdot M(H) + M(O) \)
\(M(H_2O) = 2 \cdot (1.00797) + (15.9994) = 18.01534 \; [\text{amu}]\)
Atomic and molecular weights are ratios. A mole is an amount of substance where the weight in grams is equal to the atomic or molecular weight. 1 mole of \( \ce{^12C}\) is \(12 \; [\text{g}]\). By this definition, there is a constant number of molecules in a mole, regardless of the atomic weight. This is known as Avogadro's Law where the number of atoms in a mole is given by Avogadro's number, \(N_A = 0.6022\cdot10^{24} \).
The atomic mass unit or amu is used to describe the weight of an individual atom. The mass of an atom in amu is the same as the atomic weight, since 1 amu is arbitrarily defined as \( \frac{1}{12} \cdot m(\ce{^12C})\).
The molar mass, \(MM\), is the mass in gram of one mol and is the same as the molecular mass in amu. For exmample, the atomic mass of iron is \(55.84 \; [amu]\) and the molar mass is \( 55.85 \; [\frac{g}{mol}]\). The atomic density defines how many atoms are in a unit volume:
\(N = \cfrac{N_A\rho}{MM}\)
How many grams is \(3\) mols of \(\text{Cs-}133\)?
First, we can identify the molar mass of \(\text{Cs}\) is \(132.9 \; [\text{amu}]\).
Next, we can use the formula for determining mass based on the molar mass and number of moles:
\(m = nM\)
\(m = (3 \; [\text{mol}])\!\left(132.9 \! \left[\cfrac{\text{g}}{\text{mol}}\right]\right)\)
\(m = 398.7 \; [\text{g}]\)
Therefore, we can determine there are \(398.7 [\text{g}]\) in \(3\) mols of \(\text{Cs-}133\).