<u>Answer:</u>
<u>For a:</u> The activity coefficient of copper ions is 0.676
<u>For b:</u> The activity coefficient of potassium ions is 0.851
<u>For c:</u> The activity coefficient of potassium ions is 0.794
<u>Explanation:</u>
To calculate the activity coefficient of an ion, we use the equation given by Debye and Huckel, which is:
........(1)
where,
= activity coefficient of ion
= charge of the ion
= ionic strength of solution
= diameter of the ion in nm
To calculate the ionic strength, we use the equation:
......(2)
where,
= concentration of i-th ions
= charge of i-th ions
We are given:
0.01 M NaCl solution:
Calculating the ionic strength by using equation 2:
![C_{Na^+}=0.01M\\Z_{Na^+}=+1\\C_{Cl^-}=0.01M\\Z_{Cl^-}=-1](https://tex.z-dn.net/?f=C_%7BNa%5E%2B%7D%3D0.01M%5C%5CZ_%7BNa%5E%2B%7D%3D%2B1%5C%5CC_%7BCl%5E-%7D%3D0.01M%5C%5CZ_%7BCl%5E-%7D%3D-1)
Putting values in equation 2, we get:
![\mu=\frac{1}{2}[(0.01\times (+1)^2)+(0.01\times (-1)^2)]\\\\\mu=0.01M](https://tex.z-dn.net/?f=%5Cmu%3D%5Cfrac%7B1%7D%7B2%7D%5B%280.01%5Ctimes%20%28%2B1%29%5E2%29%2B%280.01%5Ctimes%20%28-1%29%5E2%29%5D%5C%5C%5C%5C%5Cmu%3D0.01M)
Now, calculating the activity coefficient of
ion in the solution by using equation 1:
![Z_{Cu^{2+}}=2+\\\alpha_{Cu^{2+}}=0.6\text{ (known)}\\\mu=0.01M](https://tex.z-dn.net/?f=Z_%7BCu%5E%7B2%2B%7D%7D%3D2%2B%5C%5C%5Calpha_%7BCu%5E%7B2%2B%7D%7D%3D0.6%5Ctext%7B%20%20%28known%29%7D%5C%5C%5Cmu%3D0.01M)
Putting values in equation 1, we get:
![-\log\gamma_{Cu^{2+}}=\frac{0.51\times (+2)^2\times \sqrt{0.01}}{1+(3.3\times 0.6\times \sqrt{0.01})}\\\\-\log\gamma_{Cu^{2+}}=0.17\\\\\gamma_{Cu^{2+}}=10^{-0.17}\\\\\gamma_{Cu^{2+}}=0.676](https://tex.z-dn.net/?f=-%5Clog%5Cgamma_%7BCu%5E%7B2%2B%7D%7D%3D%5Cfrac%7B0.51%5Ctimes%20%28%2B2%29%5E2%5Ctimes%20%5Csqrt%7B0.01%7D%7D%7B1%2B%283.3%5Ctimes%200.6%5Ctimes%20%5Csqrt%7B0.01%7D%29%7D%5C%5C%5C%5C-%5Clog%5Cgamma_%7BCu%5E%7B2%2B%7D%7D%3D0.17%5C%5C%5C%5C%5Cgamma_%7BCu%5E%7B2%2B%7D%7D%3D10%5E%7B-0.17%7D%5C%5C%5C%5C%5Cgamma_%7BCu%5E%7B2%2B%7D%7D%3D0.676)
Hence, the activity coefficient of copper ions is 0.676
We are given:
0.025 M HCl solution:
Calculating the ionic strength by using equation 2:
![C_{H^+}=0.025M\\Z_{H^+}=+1\\C_{Cl^-}=0.025M\\Z_{Cl^-}=-1](https://tex.z-dn.net/?f=C_%7BH%5E%2B%7D%3D0.025M%5C%5CZ_%7BH%5E%2B%7D%3D%2B1%5C%5CC_%7BCl%5E-%7D%3D0.025M%5C%5CZ_%7BCl%5E-%7D%3D-1)
Putting values in equation 2, we get:
![\mu=\frac{1}{2}[(0.025\times (+1)^2)+(0.025\times (-1)^2)]\\\\\mu=0.025M](https://tex.z-dn.net/?f=%5Cmu%3D%5Cfrac%7B1%7D%7B2%7D%5B%280.025%5Ctimes%20%28%2B1%29%5E2%29%2B%280.025%5Ctimes%20%28-1%29%5E2%29%5D%5C%5C%5C%5C%5Cmu%3D0.025M)
Now, calculating the activity coefficient of
ion in the solution by using equation 1:
![Z_{K^{+}}=+1\\\alpha_{K^{+}}=0.3\text{ (known)}\\\mu=0.025M](https://tex.z-dn.net/?f=Z_%7BK%5E%7B%2B%7D%7D%3D%2B1%5C%5C%5Calpha_%7BK%5E%7B%2B%7D%7D%3D0.3%5Ctext%7B%20%20%28known%29%7D%5C%5C%5Cmu%3D0.025M)
Putting values in equation 1, we get:
![-\log\gamma_{K^{+}}=\frac{0.51\times (+1)^2\times \sqrt{0.025}}{1+(3.3\times 0.3\times \sqrt{0.025})}\\\\-\log\gamma_{K^{+}}=0.070\\\\\gamma_{K^{+}}=10^{-0.070}\\\\\gamma_{K^{+}}=0.851](https://tex.z-dn.net/?f=-%5Clog%5Cgamma_%7BK%5E%7B%2B%7D%7D%3D%5Cfrac%7B0.51%5Ctimes%20%28%2B1%29%5E2%5Ctimes%20%5Csqrt%7B0.025%7D%7D%7B1%2B%283.3%5Ctimes%200.3%5Ctimes%20%5Csqrt%7B0.025%7D%29%7D%5C%5C%5C%5C-%5Clog%5Cgamma_%7BK%5E%7B%2B%7D%7D%3D0.070%5C%5C%5C%5C%5Cgamma_%7BK%5E%7B%2B%7D%7D%3D10%5E%7B-0.070%7D%5C%5C%5C%5C%5Cgamma_%7BK%5E%7B%2B%7D%7D%3D0.851)
Hence, the activity coefficient of potassium ions is 0.851
We are given:
0.02 M
solution:
Calculating the ionic strength by using equation 2:
![C_{K^+}=(2\times 0.02)=0.04M\\Z_{K^+}=+1\\C_{SO_4^{2-}}=0.02M\\Z_{SO_4^{2-}}=-2](https://tex.z-dn.net/?f=C_%7BK%5E%2B%7D%3D%282%5Ctimes%200.02%29%3D0.04M%5C%5CZ_%7BK%5E%2B%7D%3D%2B1%5C%5CC_%7BSO_4%5E%7B2-%7D%7D%3D0.02M%5C%5CZ_%7BSO_4%5E%7B2-%7D%7D%3D-2)
Putting values in equation 2, we get:
![\mu=\frac{1}{2}[(0.04\times (+1)^2)+(0.02\times (-2)^2)]\\\\\mu=0.06M](https://tex.z-dn.net/?f=%5Cmu%3D%5Cfrac%7B1%7D%7B2%7D%5B%280.04%5Ctimes%20%28%2B1%29%5E2%29%2B%280.02%5Ctimes%20%28-2%29%5E2%29%5D%5C%5C%5C%5C%5Cmu%3D0.06M)
Now, calculating the activity coefficient of
ion in the solution by using equation 1:
![Z_{K^{+}}=+1\\\alpha_{K^{+}}=0.3\text{ (known)}\\\mu=0.06M](https://tex.z-dn.net/?f=Z_%7BK%5E%7B%2B%7D%7D%3D%2B1%5C%5C%5Calpha_%7BK%5E%7B%2B%7D%7D%3D0.3%5Ctext%7B%20%20%28known%29%7D%5C%5C%5Cmu%3D0.06M)
Putting values in equation 1, we get:
![-\log\gamma_{K^{+}}=\frac{0.51\times (+1)^2\times \sqrt{0.06}}{1+(3.3\times 0.3\times \sqrt{0.06})}\\\\-\log\gamma_{K^{+}}=0.1\\\\\gamma_{K^{+}}=10^{-0.1}\\\\\gamma_{K^{+}}=0.794](https://tex.z-dn.net/?f=-%5Clog%5Cgamma_%7BK%5E%7B%2B%7D%7D%3D%5Cfrac%7B0.51%5Ctimes%20%28%2B1%29%5E2%5Ctimes%20%5Csqrt%7B0.06%7D%7D%7B1%2B%283.3%5Ctimes%200.3%5Ctimes%20%5Csqrt%7B0.06%7D%29%7D%5C%5C%5C%5C-%5Clog%5Cgamma_%7BK%5E%7B%2B%7D%7D%3D0.1%5C%5C%5C%5C%5Cgamma_%7BK%5E%7B%2B%7D%7D%3D10%5E%7B-0.1%7D%5C%5C%5C%5C%5Cgamma_%7BK%5E%7B%2B%7D%7D%3D0.794)
Hence, the activity coefficient of potassium ions is 0.794