Calculate the ratio of degree of dissociation `((alpha_(2))/(alpha_(1)))` when 1 M acetic acid solution is diluted to 1/100 times. `["Given "K_a=10^(-5)M]`

Answer

To solve the problem of calculating the ratio of the degree of dissociation \((\frac{\alpha_2}{\alpha_1})\) when a 1 M acetic acid solution is diluted to 1/100 times, we can follow these steps:

### Step 1: Understand the Degree of Dissociation
The degree of dissociation (\(\alpha\)) is defined as the fraction of the original acid that has dissociated into ions. For a weak acid like acetic acid, the dissociation can be represented as:
\[ \text{CH}_3\text{COOH} \rightleftharpoons \text{H}^+ + \text{CH}_3\text{COO}^- \]

### Step 2: Write the Expression for \(K_a\)
The acid dissociation constant \(K_a\) for acetic acid is given by:
\[ K_a = \frac{[\text{H}^+][\text{CH}_3\text{COO}^-]}{[\text{CH}_3\text{COOH}]} \]

### Step 3: Set Up the Equilibrium Expression
Let \(C\) be the initial concentration of acetic acid. At equilibrium:
- The concentration of \(\text{H}^+\) and \(\text{CH}_3\text{COO}^-\) will be \(C\alpha\).
- The concentration of \(\text{CH}_3\text{COOH}\) will be \(C(1 - \alpha)\).

Thus, we can express \(K_a\) as:
\[ K_a = \frac{(C\alpha)(C\alpha)}{C(1 - \alpha)} = \frac{C\alpha^2}{1 - \alpha} \]

### Step 4: Approximate for Weak Acids
For weak acids, \(\alpha\) is small, so \(1 - \alpha \approx 1\). Therefore, we can simplify the expression to:
\[ K_a \approx C\alpha^2 \]

### Step 5: Calculate \(\alpha_1\) for 1 M Solution
For the initial concentration \(C_1 = 1 \, \text{M}\):
\[ K_a = 10^{-5} = 1 \cdot \alpha_1^2 \]
Thus,
\[ \alpha_1^2 = 10^{-5} \]
\[ \alpha_1 = \sqrt{10^{-5}} = 10^{-2.5} \]

### Step 6: Calculate \(\alpha_2\) for Diluted Solution
For the diluted concentration \(C_2 = \frac{1}{100} \, \text{M} = 0.01 \, \text{M}\):
\[ K_a = 10^{-5} = 0.01 \cdot \alpha_2^2 \]
Thus,
\[ \alpha_2^2 = \frac{10^{-5}}{0.01} = 10^{-3} \]
\[ \alpha_2 = \sqrt{10^{-3}} = 10^{-1.5} \]

### Step 7: Calculate the Ratio \(\frac{\alpha_2}{\alpha_1}\)
Now we can find the ratio:
\[ \frac{\alpha_2}{\alpha_1} = \frac{10^{-1.5}}{10^{-2.5}} = 10^{(2.5 - 1.5)} = 10^1 = 10 \]

### Final Answer
The ratio of the degree of dissociation when 1 M acetic acid is diluted to 1/100 times is:
\[ \frac{\alpha_2}{\alpha_1} = 10 \]