Given the following molar conductivity at `25^(@)` C:, HCl,426`Omega^(-1)cm^2mol^(-1)`, NaCl, `126Omega^(-1)cm^2mol^(-1)`, NaC(sodium crotonate), `83Omega^(_1)cm^2mol^(-1)`. What is the dissciation constant of crotonic acid,if the conductivity of a 0.001 M crotonic acid solution is `3.83xx10^(-5)Omega^(-1)cm^(-1)`?

Options

A. `10^(-5)`
B. `1.11xx10^(-5)`
C. `1.11xx10^(-4)`
D. 0.01

Answer

To find the dissociation constant of crotonic acid (K), we can follow these steps:

### Step 1: Calculate the Molar Conductivity of Crotonic Acid
The molar conductivity (λ_m) of crotonic acid (HCr) can be calculated using the formula:

\[ \lambda_m = \frac{1000 \cdot \kappa}{C} \]

Where:
- \(\kappa\) is the conductivity of the solution (given as \(3.83 \times 10^{-5} \, \Omega^{-1} \, \text{cm}^{-1}\))
- \(C\) is the concentration of the solution (given as \(0.001 \, \text{mol/L}\))

Substituting the values:

\[ \lambda_m = \frac{1000 \cdot 3.83 \times 10^{-5}}{0.001} = 38.3 \, \Omega^{-1} \, \text{cm}^2 \, \text{mol}^{-1} \]

### Step 2: Calculate the Molar Conductivity at Infinite Dilution
The molar conductivity at infinite dilution for crotonic acid can be calculated using the molar conductivities of HCl and NaCl:

\[ \lambda_m^{\infty} = \lambda_m^{\text{HCl}} + \lambda_m^{\text{NaC}} - \lambda_m^{\text{NaCl}} \]

Substituting the values:

\[ \lambda_m^{\infty} = 426 + 83 - 126 = 383 \, \Omega^{-1} \, \text{cm}^2 \, \text{mol}^{-1} \]

### Step 3: Calculate the Degree of Dissociation (α)
The degree of dissociation (α) can be calculated using the formula:

\[ \alpha = \frac{\lambda_m}{\lambda_m^{\infty}} \]

Substituting the values:

\[ \alpha = \frac{38.3}{383} \approx 0.1 \]

### Step 4: Calculate the Dissociation Constant (K)
The dissociation constant (K) for crotonic acid can be calculated using the formula:

\[ K = \frac{C \cdot \alpha^2}{1 - \alpha} \]

Substituting the values:

\[ K = \frac{0.001 \cdot (0.1)^2}{1 - 0.1} = \frac{0.001 \cdot 0.01}{0.9} = \frac{0.00001}{0.9} \approx 1.11 \times 10^{-5} \]

### Conclusion
The dissociation constant of crotonic acid is approximately \(1.11 \times 10^{-5}\).

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