Why does pH = pK when pH = 7, according to the Henderson-Hasselbalch equation?

Question

Mateo Aguilar · Accepted Answer

In fact #pH# #=# #pK_a# when #[HA]# #=# #[A^-]#; i.e. #pH# #=# #pK_a# at the point of half-equivalence.

The behavior of a weak acid in water can be represented by the following equation: #HA(aq) rightleftharpoons H^+ + A^-#. And we write the equilibrium reaction as follows, #K_a = [[H^+][A^-]]/[[HA]]# We take #log_10# of both sides: #log_10(K_a) = log_10[H^+] + log_10{[[A^-]]/[[HA]]}# After reorganization, #-log_10[H^+] = -log_10(K_a) + log_10{[[A^-]]/[[HA]]}# OR #pH = pK_a + log_10{[[A^-]]/[[HA]]}# (because by definition #-log_10[H^+] = pH#, and #-log_10(K_a)=pK_a#). I have no doubt you've seen this equation before, but what does it mean? What, precisely does the log function mean? When I write #log_ab = c#, it means to get #c# I have to raise the base (of the log) #a# to the #c^(th)# power to get #b#, i.e. #a^c =b#. Before the advent of electronic calculators, student were routinely issued log books and used to do complex mulitplications and divisions by taking logs and antilogs. Note that #log_(10)1# #=# #0#. So in the equation above, at the point of half-equivalence #[A^-]=[HA]#, and the log of this quotient #=0#. Thus at half-equivalence, #pH =pK_a#. See if you can get your head round this. In the meantime, can you tell me values for #log_(10)10#, #log_(10)100#, #log_(10)1000#, and #log_(10)1000000#? Use a calculator if you are stumped; why are these values simple whole numbers?

Owen Ambrose · Answer

undefined The Henderson-Hasselbalch equation states that pH = pKa + log([A-]/[HA]). When pH = 7, it indicates a neutral solution. In this case, [A-]/[HA] = 1, resulting in log(1) = 0. Therefore, pH = pKa when pH = 7 because log(1) equals 0.

Caleb Adams · Answer

undefined When pH = pK, according to the Henderson-Hasselbalch equation, it signifies that the concentration of the acidic and basic forms of a compound are equal. At pH 7, the solution is neutral, meaning the concentration of hydrogen ions (H⁺) equals the concentration of hydroxide ions (OH⁻), making pH equal to the pK value of the compound's acidic dissociation constant (Ka).