Answer:
a) 6.12
b) 1.87
Explanation:
At the onset of the equivalence point (i.e the first equivalence point); alaninate is being converted to alanine.
------> ![H_3}^+NC_2H_5CO^-_2](https://tex.z-dn.net/?f=H_3%7D%5E%2BNC_2H_5CO%5E-_2)
1 mole of alaninate react with 1 mole of acid to give 1 mole of alanine;
therefore 50.0 mL of 0.160 M alaninate required 50.0 mL of 0.160M HCl to reach the first equivalence point.
The concentration of alanine can be gotten via the following process as shown below;
= ![\frac{initial moles of alaninate}{total volume}](https://tex.z-dn.net/?f=%5Cfrac%7Binitial%20moles%20of%20alaninate%7D%7Btotal%20volume%7D)
= ![\frac{(50.0mL)*(0.160M)}{(50.0mL+50.0mL)}](https://tex.z-dn.net/?f=%5Cfrac%7B%2850.0mL%29%2A%280.160M%29%7D%7B%2850.0mL%2B50.0mL%29%7D)
= ![\frac{8}{100mL}](https://tex.z-dn.net/?f=%5Cfrac%7B8%7D%7B100mL%7D)
= 0.08 M
Alanine serves as an intermediary form, however the concentration of
and the pH can be determined as follows;
= ![\sqrt{\frac{K_{a1}K_{a2}{[H_3}^+NC_2H_5CO^-_2]+K_{a1}K_w}{ K_{a1}{[H_3}^+NC_2H_5CO^-_2] } }](https://tex.z-dn.net/?f=%5Csqrt%7B%5Cfrac%7BK_%7Ba1%7DK_%7Ba2%7D%7B%5BH_3%7D%5E%2BNC_2H_5CO%5E-_2%5D%2BK_%7Ba1%7DK_w%7D%7B%20%20K_%7Ba1%7D%7B%5BH_3%7D%5E%2BNC_2H_5CO%5E-_2%5D%20%20%7D%20%7D)
= ![\sqrt{\frac{ (10^{-pK_{a1})}(10^{-pK_{a2})}(0.08)+(10^{-pK_{a1})}(1.0*10^{-14})} {(10^{-pK_{a1}})+(0.08)} }](https://tex.z-dn.net/?f=%5Csqrt%7B%5Cfrac%7B%20%2810%5E%7B-pK_%7Ba1%7D%29%7D%2810%5E%7B-pK_%7Ba2%7D%29%7D%280.08%29%2B%2810%5E%7B-pK_%7Ba1%7D%29%7D%281.0%2A10%5E%7B-14%7D%29%7D%20%20%7B%2810%5E%7B-pK_%7Ba1%7D%7D%29%2B%280.08%29%7D%20%7D)
= ![\sqrt{\frac{ (10^{-2.344})(10^{-9.868})(0.08)+(10^{-2.344})(1.0*10^{-14})} {(10^{-2.344})+(0.08)} }](https://tex.z-dn.net/?f=%5Csqrt%7B%5Cfrac%7B%20%2810%5E%7B-2.344%7D%29%2810%5E%7B-9.868%7D%29%280.08%29%2B%2810%5E%7B-2.344%7D%29%281.0%2A10%5E%7B-14%7D%29%7D%20%20%7B%2810%5E%7B-2.344%7D%29%2B%280.08%29%7D%20%7D)
= ![7.63*10^{-7}M](https://tex.z-dn.net/?f=7.63%2A10%5E%7B-7%7DM)
pH = - log
pH = ![-log[7.63*10^{-7}]](https://tex.z-dn.net/?f=-log%5B7.63%2A10%5E%7B-7%7D%5D)
pH= 6.12
Therefore, the pH of the first equivalent point = 6.12
b) At the second equivalence point; all alaninate is converted into protonated alanine.
-----> ![H^+_3NC_2H_5CO^-_2](https://tex.z-dn.net/?f=H%5E%2B_3NC_2H_5CO%5E-_2)
-----> ![H^+_3NC_2H_5CO_2H](https://tex.z-dn.net/?f=H%5E%2B_3NC_2H_5CO_2H)
Here; we have a situation where 1 mole of alaninate react with 2 moles of acid to give 1 mole of protonated alanine;
Moreover, 50.0 mL of 0.160 M alaninate is needed to produce 100.0mL of 0.160 M HCl in order to achieve the second equivalence point.
Thus, the concentration of protonated alanine can be determined as:
= ![\frac{initial moles of alaninate}{total volume}](https://tex.z-dn.net/?f=%5Cfrac%7Binitial%20moles%20of%20alaninate%7D%7Btotal%20volume%7D)
= ![\frac{(50.0mL)*(0.160M)}{(50.0mL+100.0mL)}](https://tex.z-dn.net/?f=%5Cfrac%7B%2850.0mL%29%2A%280.160M%29%7D%7B%2850.0mL%2B100.0mL%29%7D)
= ![\frac{8}{150}](https://tex.z-dn.net/?f=%5Cfrac%7B8%7D%7B150%7D)
= 0.053 M
The pH at the second equivalence point can be calculated via the dissociation of protonated alanine at equilibrium which is represented as:
⇄
(0.053 - x) x x
= ![\frac{[H^+] [H^+_3NC_2H_5CO^-_2]}{[H^+_3NC_2H_5CO_2H]}](https://tex.z-dn.net/?f=%5Cfrac%7B%5BH%5E%2B%5D%20%5BH%5E%2B_3NC_2H_5CO%5E-_2%5D%7D%7B%5BH%5E%2B_3NC_2H_5CO_2H%5D%7D)
= ![\frac{x*x}{(0.053-x)}](https://tex.z-dn.net/?f=%5Cfrac%7Bx%2Ax%7D%7B%280.053-x%29%7D)
![10^{-2.344} =\frac{x^2}{(0.053-x)}](https://tex.z-dn.net/?f=10%5E%7B-2.344%7D%20%3D%5Cfrac%7Bx%5E2%7D%7B%280.053-x%29%7D)
![0.00453 = \frac{x^2}{(0.053-x)}](https://tex.z-dn.net/?f=0.00453%20%3D%20%5Cfrac%7Bx%5E2%7D%7B%280.053-x%29%7D)
![0.00453(0.053-x) =x^2](https://tex.z-dn.net/?f=0.00453%280.053-x%29%20%3Dx%5E2)
![x^2+0.00453x-(2.4009*10^{-4})](https://tex.z-dn.net/?f=x%5E2%2B0.00453x-%282.4009%2A10%5E%7B-4%7D%29)
Using quadratic equation formula;
![\frac{-b+/-\sqrt{b^2-4ac} }{2a}](https://tex.z-dn.net/?f=%5Cfrac%7B-b%2B%2F-%5Csqrt%7Bb%5E2-4ac%7D%20%7D%7B2a%7D)
we have:
OR ![\frac{-0.00453-\sqrt{(0.00453)^2-4(1)(-2.4009*10^{-4})} }{2(1)}](https://tex.z-dn.net/?f=%5Cfrac%7B-0.00453-%5Csqrt%7B%280.00453%29%5E2-4%281%29%28-2.4009%2A10%5E%7B-4%7D%29%7D%20%7D%7B2%281%29%7D)
= 0.0134 OR -0.0179
So; we go by the positive integer which says
x = 0.0134
So ![[H^+]=[H_3^+NC_2H_5CO^-_2]= 0.0134 M](https://tex.z-dn.net/?f=%5BH%5E%2B%5D%3D%5BH_3%5E%2BNC_2H_5CO%5E-_2%5D%3D%200.0134%20M)
pH = ![-log[H^+]](https://tex.z-dn.net/?f=-log%5BH%5E%2B%5D)
pH = ![-log[0.0134]](https://tex.z-dn.net/?f=-log%5B0.0134%5D)
pH = 1.87
Thus, the pH of the second equivalent point = 1.87