Category: Intermediate Chemistry

In chemical thermodynamics, the universe is composed of a system and its surroundings. The system is a part of the universe that is being studied, while its surroundings is the remaining part of the universe, with the two regions separated by a boundary. For example, if we are studying a gaseous reaction in a closed vessel in thermal equilibrium with its surroundings, the system is the gas in the vessel, while its surroundings is the space outside the vessel, and the boundary is the thermally conducting vessel walls (see diagram below).

The change in entropy of the universe is given by the sum of the change in entropy of the system and the change in entropy of the surroundings:

or under standard conditions:

The change in entropy of the universe is sometimes called the total entropy change ΔS_tot^o. However, we are mostly concerned about chemical reactions that are usually defined as occurring within a system, and especially whether such reactions are spontaneous. To determine the spontaneity of a reaction using eq3, we need to know the following facts:

• • ΔS_uni^o > 0 for a spontaneous reaction
  • ΔS_uni^o = 0 when a reaction reaches equilibrium, where there is no net transfer of heat between a system and its surroundings.

This means that we require, in addition to ΔS_sys^o, the value of ΔS_sur^o, which may not be easily available. To circumvent this problem, we seek a new measure, which can be developed as follows:  

For a reaction carried out under standard conditions, ΔS_sys^o is given by eq1 of the previous article, i.e., the standard reaction entropy of the system, ΔS_r^o, while

where T is the temperature of the surroundings, which is equal to the temperature of the system, as the system is at thermal equilibrium with its surroundings.

Although the derivation of eq4 requires knowledge of chemical thermodynamics at the advanced level, we can interpret it by, firstly, noting that the change in enthalpy of the surroundings at constant pressure is equivalent to the amount of heat transferred from the system to the surroundings. We also mentioned in the previous article that an increase in entropy of a system is associated with an increase in number of moles of substances. Rearranging eq4, we get

We can say that the ‘spread’ of a given value of ΔH_sur^o (for a reaction occurring in the surroundings) over a larger increase in the number of moles of substances (i.e. a higher value of ΔS_sur^o) leads to lower energy per particle and hence, lower T.

Having interpreted eq4, let’s consider an exothermic reaction occurring in the system that releases thermal energy that is measured as the change in enthalpy of the system, ΔH_sys^o. Since the system’s vessel walls are thermally conducting, by the theory of conservation of energy, energy transferred from the system to its surroundings increases the energy of the surroundings and decreases the energy of the system, i.e.,

Substituting eq6 in eq4,

Substituting eq7 in eq3,

From the previous article, we know that a reaction occurring in a system can result in either positive or negative ΔS_sys^o. Furthermore, we mentioned earlier that:

• • ΔS_uni^o > 0 for a spontaneous reaction
  • ΔS_uni^o = 0 when a reaction reaches equilibrium, where there is no net transfer of thermal energy between a system and its surroundings.

Therefore, with reference to eq8, if the reaction is exothermic and ΔH_sys^o is large and negative to the extent that ΔS_uni^o > 0, the reaction is spontaneous. For an endothermic reaction to be spontaneous, ΔS_sys^o must be positive enough to compensate for the term .

Substituting eq1 from the previous article and eq7 in eq3,

 

 

Eq8 is a better way to measure the spontaneity of a reaction, as it only requires the calculation of thermodynamic properties of the system. In fact, we can generate a new measure (a new function) in place of ΔS_uni^o to determine the spontaneity of a reaction, starting from:

Multiplying eq10 throughout by T,

or

where ΔG_sys^o = ΔH_sys^o – TΔS_sys^o.

We call this function, ΔG_sys^o, the Gibbs energy of the system.

Finally, just as  ΔS_uni^o  = 0 when a reaction reaches equilibrium, where there is no net transfer of thermal energy between a system and its surroundings, ΔG_sys^o = 0 for a reaction at equilibrium.

 

Next article: Gibbs energy

Previous article: Standard reaction entropy

Content page of chemical thermodynamics

Content page of intermediate chemistry

Main content page

Just as the standard reaction enthalpy is

the standard reaction entropy of a system is

where v_P and v_R are the stoichiometric coefficients of the products and reactants respectively.

The difference between ΔH_f^o and S_m^o is that the former is the relative energy between a substance and the constituents of the substance in their reference states, while the latter is the absolute energy of the substance. This is why ΔH_f^o[C(graphite)] = 0 but S_m^o[C(graphite)] ≠ 0. Despite the difference, the quantities of ΔH_r^o and ΔS_r^o are comparable, as the zero-enthalpies of reference states in and cancel out when we compute ΔH_r^o (see articles on standard enthalpy change of formation and Hess Law for details).

 

 

The positive change in standard reaction entropy of the incomplete combustion of carbon illustrates the dependency of the change in entropy of a system on the change in the number of moles of gas in a reaction. In the above example, the double in number of moles of gas outweighs the removal of a mole of carbon solid in the system, as the increase in the number of moles of gas leads to the dispersal of energy over a greater number of translational energy states of the system. This implies that ΔS_r^o can be negative when we have a reaction that results in a decrease the number of moles of gas.

 

Next article: Entropy of the universe

Previous article: Entropy & the 3rd law of thermodynamics

Content page of chemical thermodynamics

Content page of intermediate chemistry

Main content page

Entropy, denoted by the letter S, is a measure of energy dispersal at a specific temperature.

From the articles on chemical energetics, we learn that a reaction’s enthalpy change indicates whether it is exothermic or endothermic. Scientists previously thought that only exothermic reactions occur spontaneously, as the energy of systems are lowered during such reactions in a way similar to the lowering of an object’s gravitational potential energy when it falls from a height to the ground. However, the change in enthalpy of a reaction does not predict whether the reaction is spontaneous, because both exothermic and endothermic reactions can occur spontaneously. For example, both NaCl and NaOH dissolve spontaneously in water but the enthalpy change of solution of NaCl is positive, while that of NaOH is negative. 

One common aspect of the solution of NaCl and the solution of NaOH is the change of the ions from an ordered solid lattice structure to a random distribution of solvated ions throughout the vessel (see diagram above). Similarly, when ice dissolves spontaneously to form water at a temperature above 0°C, the regular arrangement of H₂O molecules in ice becomes a disordered distribution of H₂O molecules.

This suggests that chemical reactions occur spontaneously in a direction that increases their energy dispersal, which is sometimes inaccurately termed as ‘disorderliness‘. To quantify the change in the energy dispersal of a system, a measure called entropy is developed.

 

Next article: Entropy & the 3rd law of thermodynamics

Content page of chemical thermodynamics

Content page of intermediate chemistry

Main content page

A covalent solid consists of atoms held together by covalent bonds, creating a network that extends throughout the crystal, while a molecular solid is formed when molecules are held together by van der Waals forces, hydrogen bonds, or a combination of the two types of intermolecular forces.

Diamond is an example of a covalent solid with a structure similar to that of zinc blende. The structure involves carbon atoms (orange in diagram above) occupying four tetrahedral holes within a face-centred cubic structure formed by carbon atoms (blue). Like zinc blende, the unit cell of diamond is face-centred cubic. Other elements, such as silicon and germanium in group 14, also have the same structure.

An example of a molecular solid is ice. Water molecules, when cooled below 273.15 K at standard atmospheric pressure, are held together in a chair configuration (see diagram I above) by hydrogen bonds. When viewed from the top, hexagonal rings extend throughout the crystal, with the unit cell being hexagonal (demarcated by red lines in diagram II).

The structure of graphite, on the other hand, features characteristics of both a covalent solid and a molecular solid. The diagram above shows carbon atoms covalently bonded to form giant planar sheets of hexagonal rings that are stacked in an XYXY… manner. Each sheet is called graphene. Every carbon in graphite is sp² hybridised, leaving an electron in the orbital that is free to form a π-bond with one of three neighbouring orbitals. One may expect alternating single and double bonds throughout each sheet. However, x-ray diffraction data show that all covalent bonds in the sheets have a similar length of 0.141 nm, implying that the orbitals overlap to form a conjugated -electron system where the electrons are delocalised. This is the reason why graphite conducts electricity. Experiments also show that graphite is only electrically conductive in the direction parallel to the sheets, but not perpendicular to them, suggesting that the sheets are not held together by the overlap of p_z orbitals in the perpendicular direction. Layers of graphene are instead loosely stabilised by van der Waals forces and can slide against each other, making graphite a good lubricant. The thermodynamically stable form of graphite, as shown in the diagram above, has a hexagonal unit cell containing four carbon atoms.

 

 

Previous article: Radius ratio

Content page of solids

Content page of intermediate chemistry

Main content page

 

The radius ratio for an ionic solid is defined as the ratio of the radius of the smaller ion, r_s, to the radius of the larger ion r_l. An ionic solid with a 8:8 coordination structure (e.g. CsCl) has the following radius ratio:

where a is the length of a side of the cube. Using Pythagoras’ theorem, the body diagonal of the cube, d, is:

Substituting eq1 in eq2

The body diagonal is also given by:

Substituting eq3 in eq4 and rearranging,

The radius ratio of 0.732 is the minimum ratio required for a 8:8 coordination compound. If the radius of the cation is smaller, it will not be in contact with the anions, resulting in a compound that has a coordination of less than 8:8, i.e. a 6:6 coordination compound.

The diagram above shows one of the six surfaces of an ionic solid with a 6:6 coordination structure (e.g. NaCl). Its radius ratio is calculated as follows:

where f is the face diagonal. Substituting eq5 in eq6,

The face diagonal is also given by:

Combining eq7 and eq8 and simplifying,

Again, the radius ratio of 0.414 is the minimum ratio required for a 6:6 coordination compound. If the radius of the cation is smaller, it will not be in contact with the anions, resulting in a compound that has a coordination of less than 6:6, i.e. a 4:4 coordination compound.

Finally, for a 4:4 coordination structure like zinc blende (see diagram above), its radius ratio is calculated by selecting the appropriate ions (diagram IIf):

Since the tetrahedral angle is 109.5°,

Substituting eq10 in eq11 and simplifying,

Once again, the radius ratio of 0.225 is the minimum ratio required for a 4:4 coordination compound.

Therefore, assuming that ions in crystal are rigid spheres and that cations are in contact with anions, we can use the three calculated radius ratios to predict structures of ionic compounds as follows:-

Radius Ratio
Coordination	4:4	6:6	8:8
Structure	Zinc blende type	NaCl type	CsCl type

 

Next article: Covalent and molecular solids

Previous article: ionic solids

Content page of solids

Content page of intermediate chemistry

Main content page

Ionic solids are characterised by robust lattice structures formed through the electrostatic attraction between positively and negatively charged ions.

We can, as we did for metallic solids, use the concept of packing spheres to study ionic solids. We also need to consider the relative sizes of the oppositely charged ions and the charges of the ions to determine the structure.

Consider two layers of closely packed anions, with the second layer (orange) sitting in the notches of the bottom layer (blue), forming two types of space (or holes) between the layers (see diagram I above). As the volume of a cation is usually smaller than that of an anion, a cation can fit into either hole, depending on the relative sizes of cations and anions in the crystal. A relatively larger cation occupies an octahedral hole, while a relatively smaller cation lodges in a tetrahedral hole (diagram II), forming ionic compounds with two different arrangements. For each of the arrangements, the anion-cation-anion layered structure then repeats itself with the third layer of anions having the same arrangement as the first layer of anions.

For the ionic compound where octahedral holes are filled, each cation has six anions in its vicinity and each anion has six cations in its vicinity. Therefore, we also call such a structure a 6-6 coordination structure. For the ionic compound where tetrahedral holes are filled, each cation has four anions in its vicinity and each anion has four cations in its vicinity. Therefore, we also call such a structure a 4-4 coordination structure.

If the cation-anion radius ratio is even larger (but still less than 1), the layer of anions must move apart to accommodate the cations (diagram III), such that both cations and anions are in the same layer with every cation in contact with four anions (diagram IV). A second layer has the anions sitting in the notches formed by the anions of the bottom layer (diagram V), with each second-layer cation again in contact with four second-layer anions. The third layer of anions are directly above those in the first layer, giving the crystal an XYXY… structure. In such a structure, each cation has eight anions in its vicinity and each anion has eight cations in its vicinity. Therefore, we also call such a structure a 8-8 coordination structure.

Having seen how cations and anions are packed in a crystal, let’s find the respective unit cells. Firstly, we depict a crystal that has a packing pattern like diagram V in the lattice form (diagram Va), and rotate it for a better view (diagram Vb). Note that some anions are coloured black for contrast.

The space lattice Vb is formed by replicating two interpenetrating cells, a cationic simple cubic unit cell (pink) and an anionic simple cubic unit cell (blue; see diagram Vc). By pairing a cation in the middle of an anionic simple cubic unit cell with one of the anions of that cell, and using that as a basis (diagram Vd), we get the unit cell of V, which is again a simple cubic unit cell (diagram Ve). To maintain charge neutrality within the crystal, the total charges for cations and anions must be equal and opposite. In short, a compound with a relatively high radius ratio like caesium chloride adopts this structure.

A compound with a lower radius ratio (e.g. NaCl) has cations occupying octahedral holes that are formed by layers of anions (see diagram II and IIa below). The result is a space lattice consisting of replications of two interpenetrating cells, a cationic face-centred cubic unit cell and an anionic face-centred cubic unit cell (diagram IIb). By pairing a cation with an anion and using the pair as a basis, we get a face-centred cubic unit cell for NaCl.

For a compound with even lower radius ratio (e.g. ZnS, zinc blende), cations occupy tetrahedral holes formed by layers of anions (see diagram II and IId below). By pairing a cation with an anion and using the pair as a basis, we once again get a face-centred cubic unit cell for zinc blende (diagram IIe).

 

Next article: Radius ratio

Previous article: Metallic solids

Content page of solids

Content page of intermediate chemistry

Main content page

The crystal structures of metallic solids are usually composed of three types of unit cells that are formed due to the way metal atoms are packed in a crystal.

Consider the atoms as hard spheres touching each other to form a closely packed single layer (see diagram above). To maintain the close packed arrangement, a second layer (orange) is placed above the bottom layer (blue) with each second-layer atom sitting in the notches of the bottom layer. There are two ways to position atoms for the third layer (yellow), starting with notch A or notch B. If the first atom in the third layer starts at notch A, all atoms of the third layer are directly above atoms in the first layer, with the layers having an XYXYXY… pattern, giving the crystal (e.g. magnesium) a structure called hexagonal close-packed (hcp). However, if the first atom in the third layer starts at notch B, all atoms of the third layer are staggered relative to atoms in the first layer, resulting in the layers having an XYZXYZXYZ… pattern. A crystal with such an arrangement of layers (e.g. copper) has a structure called cubic close-packed (ccp).

Another common crystal structure of metals, shown in the diagram below, is the body centred cubic (bcc). The bottom layer (blue) is less closely packed than hcp and ccp, with each atom touching only four neighbouring atoms in the same layer instead of six. The second layer of atoms (orange) sits in the notches of the bottom layer and the third layer fits into the notches of the second layer, giving an XYXYXY… pattern. Iron and potassium have bcc structures.

Having seen how metal atoms are packed in a crystal, let’s find the respective unit cells. A crystal with a ccp structure has a face-centred cubic unit cell (see below diagram) where a = b = c and α = β = γ = 90°. There is one lattice point at each corner of the cube and one lattice point in the middle of each surface plane. Each lattice point meets the criteria of having the same environment by having six and eight neighbouring atoms at distances a and (√2/2)a respectively (it is easier to visualise the neighbouring atoms by replicating the unit cell around itself).

A crystal with a bcc structure has a unit cell of the same name (see below diagram) where a = b = c and α = β = γ = 90°. There is one lattice point at each corner of the cube and one lattice point in the middle of the cube. Each lattice point meets the criteria of having the same environment by having six and eight neighbouring atoms at distances a and (√3/2)a respectively (using Pythagoras’ theorem).

To map out the unit cell of a hcp structure, we let a pair of atoms be a lattice point (denoted by a yellow oval in the 3rd diagram below). The atom pair or basis consists of an atom from the bottom layer (blue) and one from the second layer (orange). Connecting the centres of four of the ovals (numbered 1 to 4), we get a rhombus with two opposite angles of 120°; and linking two adjacent rhombi, we obtain the unit cell called the hexagonal unit cell. The reason it is called a hexagonal unit cell is the combination of three such unit cells form a hexagon.

Finally, the different forms of packing and hence, unit cells, are a result of the different electron densities of atoms and molecules and their interactions to produce energetically stable arrangements in the solid state.

 

 

Next article: Ionic solids

Previous article: The unit cell

Content page of solids

Content page of intermediate chemistry

Main content page

The unit cell is a parallelepiped that is the simplest repeating unit of a crystal (diagram I).

Diagram II shows a crystal that is composed of unit cells where a = b = c and α = β = γ = 90°. Each blue point, known as a lattice point, must have identical surroundings (or environment) in the crystal. An infinite three-dimensional array of lattice points forms a space lattice. The unit cell in II that has eight lattice points at all corners of a cube is called a simple cubic unit cell.

Consider a crystal structure containing two types of atoms (diagram III). As each lattice point must have the same surroundings, it is, in this case, a point consisting of a pair of blue and red atoms (diagram IV). Therefore, each lattice point, in general, may represent an atom, a molecule or a collection of atoms or molecules. Using the pair of atoms as a basis in forming the space lattice, we find that the unit cell for compound III is also a simple cubic unit cell.

Finally, a unit cell must fill all space of the crystal when replicated. As a result, spherical unit cells do not exist.

 

Next article: metallic solids

Content page of solids

Content page of intermediate chemistry

Main content page

The nuclear binding energy per nucleon of a nucleus is the total binding energy of the nucleus divided by the total number of nucleons.

⁵⁶Fe has the highest nuclear binding energy per nucleon and is therefore the most stable isotope (see graph above). The negative gradient of the curve beyond ⁵⁶Fe implies that electrostatic forces of repulsion increase per nucleon for isotopes with mass number greater than 56.

 

 

previous article: mass defect

Content page of mass in chemistry

Content page of intermediate chemistry

Main content page

Mass defect is the difference in the combined masses of an atom’s components and the measured mass of the atom.

An atom is composed of electrons and nucleons (protons and neutrons). The constituents of its nucleus, the neutrons and positively charged protons, are held together by an energy called the nuclear binding energy, E_binding. In other words, the nuclear binding energy is the energy that is released when a stable nucleus is formed from its components. The total energy of the atom can be expressed as:

This equation represents the balance of energies within the atom, where E_binding accounts for the energy required to hold the nucleus together. As energy and mass are related by Einstein’s formula of E = mc², we can rewrite eq8 as:

Eq9 shows that the mass of an atom is always less than the sum of the masses of its isolated components, with the exception of ¹H where m_neutrons = 0 and thus m_binding = 0. As this defect in mass (for atoms other than ¹H) is due to the atom’s nuclear binding energy, we refer the mass equivalent of the nuclear binding energy as the mass defect of the atom:

Hence, mass defect is the difference in the combined masses of an atom’s components and the measured mass of the atom.

The heavier the atom, that is the more protons and neutrons it has, the greater the mass defect, since a larger amount of binding energy is needed to hold a greater number of positively charged protons and the neutrons together.

As an illustration, if we simply add the subatomic particles of a deuterium atom, we have:

Substituting the values of subatomic masses from the previous article in the above equation, we have: m_{D, calculated} = 2.01648996 u. The value of the relative mass of deuterium that is experimentally determined via mass spectrometry is 2.014104 u. The difference of about 0.00238596 u is due to the mass defect of deuterium.

 

NExt article: Nuclear binding energy per nucleon

previous article: Subatomic particle mass

Content page of mass in chemistry

Content page of intermediate chemistry

Main content page