Definitions

Arrhenius theory of acids and bases

• acid is a compound that can release a proton $H^{+}$ in an aqueous solution
• base is a compound that can release a hydroxide ion $O{H}^{-}$

Brønsted-Lowry theory of acids and bases

• acid is a compound that can release a proton $H^{+}$ in an aqueous solution
  • the water molecules immediately bonds with the lone proton via a donor-acceptor bond, which produces a hydronium ion $H_3{O}^{+}$

$$HA+H_{2}O\leftrightharpoons A^{-}+H_3{O}^{+}$$ - solutions with an increased concentration of the hydronium ion are called acidic

• base is a compound that can accept a proton $H^{+}$ in an aqueous solution
  • in a solution of just the base, the proton is provided by a water molecule and is bonded to the base via a donor-acceptor bond, which produces a hydroxide ion

$$B+H_{2}O\leftrightharpoons B{H}^{+}+O{H}^{-}$$ - solutions with an increased concentration of the hydroxide ion are called basic

Self-ioisation of water

• water can act as both an acid and a base
• it reacts with itself to create both a hydronium ion and a hydroxide anion

$$H_{2}O+H_{2}O\leftrightharpoons O{H}^{-}+H_3{O}^{+}$$ - the equilibrium constant for this reaction $K_w$ can be calculated as $K_w=[H_3{O}^{+}][O{H}^{-}]$ - at room temperature $K_w={10}^{-14}$ - the constant is temperature dependent - it is also called the ionic product of water

pH and pOH scales

• concentration of $H_3{O}^{+}$ and $O{H}^{-}$ is equal to ${10}^{-7}$ mol
• the pH pOH scale is defined as the negative decadic logarithm of relevant concentrations of $H_3{O}^{+}$ and $O{H}^{-}$

$$pH=-\log [H_3{O}^{+}]$$ $$pOH=-\log [O{H}^{-}]$$

• pH is commonly used to describe acidity of a solution
  • when pH is equal to 7, the solution is neutral
    • $pH=7=pOH$
  • when pH is less then 7, the solution is acidic
    • $pH<7<pOH$
  • when pH is more than 7, the solution is basic
    • $pH>7>pOH$

pH calculation

Strong acids and bases

• the equilibrium constant for the dissociation reactions of strong acids and bases is very high
  • most of the acid or base dissociates
• the pH is completely determined by the initial concentration of the acid (or base)

$$pH=-\log c_{HA}$$ $$pOH=-\log c_B$$ $$pH=pKw-pOH$$

Weak acids

$$K_a=\frac{[H_3{O}^{+}][A^{-}]}{[HA]}$$

• simplifications:
  • $[H_3{O}^{+}]=[A^{-}]$ (both are created at the same rate)
  • $c_{HA}=[HA]+[A^{-}]$
  • $[HA]=c_{HA}-[H_3{O}^{+}]\approx c_{HA}$ (assuming only small ammount of acid is dissociated)

$$K_a=\frac{[H_3{O}^{+}{]}^2}{c_{HA}}$$ $$[H_3{O}^{+}]=\sqrt{c_{HA}{K}_a}$$ $$pH=-\frac{-log{c}_{HA}+log{K}_a}{2}$$ $$pH=\frac{pKa-log{c}_{HA}}{2}$$

Ionisation fraction $x$

• it determines what precentage of the acid dissociates

$$x=\frac{[H_3{O}^{+}]}{c_{HA}}$$

• after substitution to the $K_a$ formula:

$$x=\sqrt{\frac{K_a}{c_{HA}}}$$

Weak bases

• similar process to the one used with weak acids can be used to calculate pOH of weak bases

$$pOH=\frac{p{K}_b-log{c}_B}{2}$$

• pH can be calculated using this equation:

$$K_a{K}_b=K_w$$ $$p{K}_a+p{K}_b=pKw$$ $$pH=pKw-pOH$$

• after substitution:

$$pH=pKw-\frac{p{K}_b-log{c}_B}{2}$$ $$pH=pKw-\frac{(p{K}_w-p{K}_a)-log{c}_B}{2}$$ $$pH=\frac{p{K}_w+p{K}_a+\log c_B}{2}$$

Ionisation fraction $x$

$$x=\frac{[O{H}^{-}]}{c_B}$$ $$x=\sqrt{\frac{K_b}{c_B}}$$

Acid-base reactions buffers

Strong acid and strong base

• a stong acid dissociates almost completely and releases a proton
• a strong base (usually a hydroxide) releases a hydroxide anion
• these two ions are in an equilibrium in water and thus together form water

$$O{H}^{-}+H_3{O}^{+}⟶2H_{2}O$$

• the pH is determined by the ammount of the acid and hydroxide added to the reaction
  • if the same amount of protons and hydroxide anions is added, the overall pH is neutral
  • if surplus of acid is added, the solution will be acidic, the pH can be calculated from the concentraion of the excess acid
  • if surplus of base is added, the solution will be basic, the pH can be calculated from the concentration of the excess base

Weak acid and strong base

• a weak acid only reacts with the strong base after the base yealds an $O{H}^{-}$ group
• if a surplus of base is added, all acid is neutralized and the pH will be basic
• if a surplus of acid is added, the base determines how much of it will be neutralized, but the pH is still going to be basic
  • this is because the neutralization reacton of the acid exists in an quilibrium, not in an irreversible reaction

$$HA+O{H}^{-}\leftrightharpoons A^{-}+H_{2}O$$

Strong acid and weak base

• the case is analogous to the weak acid and strong base
• the resulting solution will be acidic

$$B+H_3{O}^{+}\leftrightharpoons B{H}^{+}+H_{2}O$$

Buffers

• buffer is a solution which regulates the overall pH of a solution
• they are a result of either:
  • a solution of a weak acid with half the concentration of a strong base (compared to the concentration of the weak acid)
    • $pH\approx p{K}_a$
    • $p{K}_a\approx [H_3{O}^{+}]$
  • a solution of a weak base with half the concentration of a strong acid
    • $pH\approx p{K}_b$
    • $p{K}_b\approx [O{H}^{-}]$
• their pH doesn’t change when small ammounts of acids or bases are added
  • it initially does, but is immediately reverted

pH calculation

• pH is calculated using the Henderson-Hasselbalch equation

$$pH=p{K}_a+\log \frac{[A^{-}]}{[HA]}$$

• it’s derived from the equation for acidity constant equation
  • $K_a=\frac{[H_3{O}^{+}][A^{-}]}{[HA]}$
• as long as the fraction $\frac{[A^{-}]}{[HA]}$ stays fixed, the actual concentrations don’t play a role on the pH
  • the actual concentrations play a role however in the capacity of the buffer for adding either acids or bases
    • the higher the $[A^{-}]$ the more acid can be added without significantly affecting the pH
    • the higher the $[HA]$ the more base can be added without significantly affecting the pH
  • the ration shouldn’t be different by the factor of 10 for the buffer to work well
    • however, it is not a universal rule

Weak acid and weak base

• they react according to this equation:

$$HA+B\leftrightharpoons A^{-}+B{H}^{+}$$

• the pH calculation is rather more complex and is dependent on the exact balance of $K_a$ and $K_b$

Acid-base titration

• acid-base titrations is a method of analytical chemistry based on neutralisation of acidic (alkalimetry) or basic (acidimetry) solutions
• pH indicators are added to the analysis solutions to determine the point of equivalence
  • the solution reaches neutral pH
• titration curve plots the pH value against the volume of acid added
  • pH is reduced rapidly at the point of equivalence
  • titration curves of polyprotic acids show multiple jumps of pH

Strength of acids and bases

• factors contributing to strength of an acid
  • the bond between the acid and the acidic hydrogen is strongly polar
    • hence, the atom the hydrogen is bonded to is very electronegative
  • the bond between the acid and the acidic hydrogen is weak
• the rest of the acid’s structure also influences its ability to release a proton
  • for inorganic acids:
    • the higher the ratios between hydrogens and oxygens, the easier for it it is to release a proton
    • in hydrohalogenic acids, the strength decreases from chlorine to iodine
      • hydrofluoric acid is a weak acid (the bond between hydrogen and fluorine is too strong)
  • for organic acids:
    • they are generaly weaker than most inorgnanic acids
    • the shorter the hydrocarbon chain, the stronger the acid
  • for polyprotic acids:
    • already deprotonated acids are more difficult to further deprotonate
• strong bases usually contain an ionically bonden $-OH$ group
  • these are usually hydroxides
• bases that contain free electron pairs are generally weaker
  • they can bond a proton via a donor-acceptor bond