Respuesta :
Answer:
For a: The activity coefficient of copper ions is 0.676
For b: The activity coefficient of potassium ions is 0.851
For c: The activity coefficient of potassium ions is 0.794
Explanation:
To calculate the activity coefficient of an ion, we use the equation given by Debye and Huckel, which is:
[tex]-\log\gamma_i=\frac{0.51\times Z_i^2\times \sqrt{\mu}}{1+(3.3\times \alpha _i\times \sqrt{\mu})}[/tex] ........(1)
where,
[tex]\gamma_i[/tex] = activity coefficient of ion
[tex]Z_i[/tex] = charge of the ion
[tex]\mu[/tex] = ionic strength of solution
[tex]\alpha _i[/tex] = diameter of the ion in nm
To calculate the ionic strength, we use the equation:
[tex]\mu=\frac{1}{2}\sum_{i=1}^n(C_iZ_i^2)[/tex] ......(2)
where,
[tex]C_i[/tex] = concentration of i-th ions
[tex]Z_i[/tex] = charge of i-th ions
- For a:
We are given:
0.01 M NaCl solution:
Calculating the ionic strength by using equation 2:
[tex]C_{Na^+}=0.01M\\Z_{Na^+}=+1\\C_{Cl^-}=0.01M\\Z_{Cl^-}=-1[/tex]
Putting values in equation 2, we get:
[tex]\mu=\frac{1}{2}[(0.01\times (+1)^2)+(0.01\times (-1)^2)]\\\\\mu=0.01M[/tex]
Now, calculating the activity coefficient of [tex]Cu^{2+}[/tex] ion in the solution by using equation 1:
[tex]Z_{Cu^{2+}}=2+\\\alpha_{Cu^{2+}}=0.6\text{ (known)}\\\mu=0.01M[/tex]
Putting values in equation 1, we get:
[tex]-\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[/tex]
Hence, the activity coefficient of copper ions is 0.676
- For b:
We are given:
0.025 M HCl solution:
Calculating the ionic strength by using equation 2:
[tex]C_{H^+}=0.025M\\Z_{H^+}=+1\\C_{Cl^-}=0.025M\\Z_{Cl^-}=-1[/tex]
Putting values in equation 2, we get:
[tex]\mu=\frac{1}{2}[(0.025\times (+1)^2)+(0.025\times (-1)^2)]\\\\\mu=0.025M[/tex]
Now, calculating the activity coefficient of [tex]K^{+}[/tex] ion in the solution by using equation 1:
[tex]Z_{K^{+}}=+1\\\alpha_{K^{+}}=0.3\text{ (known)}\\\mu=0.025M[/tex]
Putting values in equation 1, we get:
[tex]-\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[/tex]
Hence, the activity coefficient of potassium ions is 0.851
- For c:
We are given:
0.02 M [tex]K_2SO_4[/tex] solution:
Calculating the ionic strength by using equation 2:
[tex]C_{K^+}=(2\times 0.02)=0.04M\\Z_{K^+}=+1\\C_{SO_4^{2-}}=0.02M\\Z_{SO_4^{2-}}=-2[/tex]
Putting values in equation 2, we get:
[tex]\mu=\frac{1}{2}[(0.04\times (+1)^2)+(0.02\times (-2)^2)]\\\\\mu=0.06M[/tex]
Now, calculating the activity coefficient of [tex]K^{+}[/tex] ion in the solution by using equation 1:
[tex]Z_{K^{+}}=+1\\\alpha_{K^{+}}=0.3\text{ (known)}\\\mu=0.06M[/tex]
Putting values in equation 1, we get:
[tex]-\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[/tex]
Hence, the activity coefficient of potassium ions is 0.794
