Extent of solubility - Entropy: how far?

Extent of solubility

·         Calculation of enthalpy of solution:

The enthalpy of solution is calculated according to that equation:

ΔH _solution =  -ΔH _lattice + ΔH_hyd (X⁺) + ΔH_hyd (Y^-)

However, extra care must be taken with ionic compounds of formula MX₂. The hydration enthalpy of the anion must be multiplied by two, because there’re two anions in the compound:
M²⁺ and 2X^-.

·         Extent of solubility of solid

Solubility of a solid is determined by the total entropy change ΔS° _(total). For a solid to dissolve ΔS° _(total) must be +ve. Its value depends on both the ΔS° _(system) and ΔH _solution.

 ΔS° _(total) = ΔS° _(system) + ΔS° _{(surroundings)} = ΔS° _(system)  + (-ΔH _solution/T)

-          Entropy of the surroundings ΔS° _{(surroundings)}

 For endothermic reactions ΔS° _{(surroundings)} is negative. Therefore, for the ionic solid to be soluble ΔS° _(system) must be positive and must outweigh the negative ΔS° _{(surroundings)}. Insoluble solids have a negative ΔS° _(total). That means that the ΔS° _(system) has a smaller positive value than the value of ΔS° _{(surroundings)} or is negative.

  The same concept applies to exothermic dissolving of solids.

-          Entropy change of the system

The entropy of the system is made up of the entropy change of the solute and the entropy change of the solvent.

ΔS° _(system) = ΔS° _(solute) + ΔS° _(solvent)

Entropy of the solute ΔS° _(solute) is ALWAYS positive and increases when the solute’s dissolved, as the particles go from the state of being arranged in a regular pattern to being distributed randomly in solution.

However, the entropy of the solvent ΔS° _(solvent) could be either +ve or –ve.

It is positive when the solvent mixes with the solute particles, i.e. more disorder

It is negative when the water molecules surround the ions of the anhydrous ionic compound, i.e. water molecules become more ordered.

The δ^- oxygen atoms in the water bond with the positive ions causing the δ⁺ hydrogen atoms of the same molecule to be more positive hence they bind a second sphere of water molecules.  

The extent to which this (hydration) happens depends on the charge density of the cation. The greater the charge density of the cation, the more ordered the water molecules are.

Cations with large charge density have a greater positive charge and smaller ionic radius.

For instance, lithium ion Li⁺ decreases the entropy of the solvent water (more ordered) to a greater extent than the other group 1 ions; this is because Li⁺ has the smallest ionic radius of all group 1 ions. That means Li⁺ has the greatest charge density of all group 1 ions.

 Ammonium ion NH₄⁺ has smaller charge density than Li⁺ because the positive charge of NH₄⁺ is spread over the whole ion (which has a much larger ionic radius). Therefore, the entropy of the solvent water is not affected as much (less ordered).

Therefore we assume that dissolving group 1 compounds and ammonium compounds causes only a small change in the entropy of water, so ΔS° _(system) is always positive.

Group 2 cations have greater positive charge. Their ionic radius is much less than of the group 1 ions in the same period. So they have a greater charge density, causing a larger decrease in the entropy of the water. Therefore, we assume that dissolving group 2 compounds results in negative ΔS° _(system).

Lattice Energy & Hydration Energy - Entropy: how far?

Factors that affect lattice energy:

The magnitude of lattice energy depends on the forces acting on the ions. In a lattice, each ion is surrounded by a number of ions of opposite charge, resulting in strong forces of attraction and some forces of repulsion.

According to Coulomb’s law [F=kQ₁Q₂/r²] where Q=charge on ion and r= sum of the radii

The strength of these forces of attraction and repulsion depend on:

·         The product of the charges on the ions – the greater the product of the charges, the greater the forces and the more exothermic the lattice energy

·         The sum of the radii of the cation and the anion – the larger the sum, the smaller the forces hence the less exothermic the lattice energy

·         The extent of covalency – the greater the extent of covalency, the greater the forces and the more exothermic the lattice energy

Example question: Explain why the lattice energy of NaF is more exothermic than the lattice energy of KCl.

 First of all, identify the differences and similarities between these two compounds.

The ionic lattice of NaF contains the ions: Na⁺ and F^-

The ionic lattice of KCl contains the ions:  K⁺   and Cl^-

Similarities: The product of the charges is the same for both compounds.

Differences: Na⁺ has as a smaller ionic radius than K⁺ and F^- is smaller than Cl^-.

What is the conclusion?

Therefore, the forces between the sodium ions and the fluoride ions are stronger than those between potassium ions and chloride ions, so the lattice energy for NaF is more exothermic than for KCl.

Lattice energy/ kJ mol -1
Halides	LiCl                -846	NaCl                    -771
Oxides	Li2O               -2814	Na2O                   -2478
	BeO               -4444	MgO                   -3890
Sulfates	Li2S                -2500	BeS                     -3832
Hydroxides		Mg(OH)2                   -2842

§  The magnitude of the lattice energy steadily decreases (becomes less exothermic; less heat given off) down a group of the periodic table as the size of the cation increase

§  The magnitude of the lattice energy steadily decreases down the group as the size of the anion increases

§  The magnitude of the lattice increases (becomes more exothermic; more energy released) as the charge on either or both the cation and the anion increases.

Factors that affect hydration energy:

Similar to lattice energy, forces between the ions and water molecules surrounding then depend on:

§  Charge on the ion

The greater the charge on the ion, the greater is the force. The hydration energies become more exothermic as the charge on the ion increases.

§  The radius of the ions

The smaller the radius, the greater is the force. The hydration energies become less exothermic as the radius of the ions in a group increases.

Definitions you need to know - Solubility -Entropy: how far?

Lattice Energy is the enthalpy change (energy released) when one mole of an ionic solid is formed from its constituent gaseous ions infinitely far apart.

X^a+ _(g) +Y^b- _(g) → X_bY_{a (s)}   ΔH _lattice

Enthalpy change of hydration is the enthalpy change (energy released) when one mole of gaseous ions dissolved in sufficient solvent to given infinity dilute solution.

X^a+ _(g) + aq → X^a+ _(aq)     ΔH_hyd (X^a+)

Y^b- _(g) + aq→ Y^b- _(aq)      ΔH_hyd (Y^b-)

Enthalpy change of solution is the enthalpy change when one mole of a compound is dissolved in sufficient solvent to give an infinitely dilute solvent.

X_bY_{a (s)} + aq→ X_bY_{a (aq)}  

ΔH _solution =  -ΔH _lattice + ΔH_hyd (X^a+) + ΔH_hyd (Y^b-)

An Infinitely dilute solvent is a solution in which further dilution does not cause a heat change.

Solubility of ionic compounds - Entropy: how far?

Solubility of ionic compounds

When an ionic solid dissolves, the following must take place:

STEP 1: The ionic lattice breaks; ions are separated.

For example:                      NaCl _(s) → Na⁺ _(g) + Cl^- _(g)   ΔH positive

Note that the reaction above is the reverse reaction of:
Na⁺ _(g) + Cl^- _(g) → NaCl _(s) where ΔH negative = ΔH _lattice

(see link for definition of lattice energy)

Therefore, ΔH positive of the reaction at the top equals to   - ΔH _lattice

STEP 2:  The positive cation forms dipole-ion forces with δ^-O of the water molecules and negative anions with δ⁺H atoms of the water molecules. This process is called Hydration of ions, which is exothermic. (see link  for definition of enthalpy of hydration)

Hydration of ions

For example:           Na⁺ _(g) + Cl^- _(g)   → Na⁺ _(aq) + Cl^- _(aq)   

The above reaction is divided into two parts:
Na⁺ _(g) + aq   → Na⁺ _(aq)   ΔH_hyd (Na⁺)
and
Cl^- _(g)  + aq → Cl^- _(aq)       ΔH_hyd (Cl^-)

On addition, the result reaction is:

 

NaCl _(s) → Na⁺ _(aq) + Cl^- _(aq)        OR      NaCl _(s) + aq → NaCl _(aq)    ΔH _solution

 
Therefore  

ΔH _solution =  -ΔH _lattice + ΔH_hyd (Na⁺) + ΔH_hyd (Cl^-)

 From this equation, it can be seen that:  

·         The more exothermic the lattice energy, the more endothermic (or less exothermic) the enthalpy of solution

·         The more exothermic either of the hydration enthalpies, the more exothermic the enthalpy of solution

 

Exothermic dissolving occurs when: the magnitude of the lattice enthalpy is less than the sum of the hydration enthalpies of the two ions.

Endothermic dissolving occurs when: the magnitude of the lattice enthalpy is greater than the sum of the hydration enthalpies of the two ions.

See the following links for more information on:

 Factors affecting Lattice Energy and Hydration Energy

Introduction to Solubility - Entropy: how far?

Solubility

Dissolving a gas: X _(g) ⇌ X _(aq)


Dissolving a gas always results in a negative ΔS° _(system) because the system becomes more ordered.

Since ΔS° _(total) = ΔS° _(system) + ΔS° _{(surroundings).}  For a gas to be soluble, the above reaction must be thermodynamically spontaneous, i.e. ΔS° _(total) must be +ve. Therefore, ΔS° _{(surroundings)} has to be +ve in order to compensate for the –ve value of ΔS° _(system). Exothermic reactions have a +ve ΔS° _{(surroundings).}
∴ Gasses dissolve exothermically.

For example:  CO_{2 (g)} ⇌   CO_{2 (aq)}  ΔH negative
The above equilibrium is driven to the left, the endothermic side, by an increase in temperature.  Hence, gases such as carbon dioxide are more soluble in cold water than hot water.

Note: Most gases have a negative enthalpy of solution (ΔH _solution). A negative enthalpy of solution means that the solute is less soluble at high temperatures.



Dissolving a solid: X _(s) ⇌ X _(aq)

Contrary to the expected approach to entropy, dissolving solids does not always result in +ve  ΔS° _(system); the system becoming more disordered.  The solute (e.g. salt) becomes more disordered as it goes from a highly ordered solid to a more random solution, however the solvent (e.g. water) can become more ordered due to the forces of attraction between the solute and the solvent. This is more likely to happen when compounds containing ions of high charge density dissolve in water.

Note: treat ΔH _solution as ΔH _reaction; therefore, if ΔH _solution is –ve it means that the reaction is exothermic. If ΔH _solution is +ve then the reaction is endothermic.

Therefore for this reaction X _(s) ⇌ X _(aq):

ΔH	ΔS° (surroundings)
+ve (endothermic)	-ve (less disordered)
-ve (exothermic)	+ve (more disordered)

Thus, at higher temperatures, the equilibrium: X _(s) ⇌ X _(aq):

is driven to the left (less soluble) if ΔH _solution is exothermic and to the right (more soluble) if ΔH_solution is endothermic.