The hydrogen ion concentration in weak acid of dissociation constant `K_(a)` and concentration `c` is nearly equal to

Options

A. `sqrt(K_(a)/C)`
B. `C/K_(a)`
C. `K_(a)C`
D. `sqrt(CK_(a))`

Answer

To find the hydrogen ion concentration in a weak acid with a given dissociation constant \( K_a \) and concentration \( c \), we can follow these steps:

### Step 1: Understand the dissociation of the weak acid
Consider a weak acid \( HA \) that dissociates in water:
\[ HA \rightleftharpoons H^+ + A^- \]

### Step 2: Set up the initial concentrations
Initially, the concentration of the weak acid is \( c \), and the concentrations of the ions \( H^+ \) and \( A^- \) are both 0.

### Step 3: Define the change in concentration
Let \( \alpha \) be the degree of dissociation of the acid. After dissociation, the concentrations can be expressed as:
- Concentration of \( HA \): \( c - c\alpha = c(1 - \alpha) \)
- Concentration of \( H^+ \): \( c\alpha \)
- Concentration of \( A^- \): \( c\alpha \)

### Step 4: Write the expression for the dissociation constant \( K_a \)
The acid dissociation constant \( K_a \) is given by:
\[ K_a = \frac{[H^+][A^-]}{[HA]} \]
Substituting the concentrations we have:
\[ K_a = \frac{(c\alpha)(c\alpha)}{c(1 - \alpha)} \]
This simplifies to:
\[ K_a = \frac{c^2 \alpha^2}{c(1 - \alpha)} \]
\[ K_a = \frac{c \alpha^2}{1 - \alpha} \]

### Step 5: Assume \( \alpha \) is small
For weak acids, we can assume that \( \alpha \) is much less than 1, so \( 1 - \alpha \approx 1 \). Thus, we can simplify the equation:
\[ K_a \approx c \alpha^2 \]

### Step 6: Solve for \( \alpha \)
Rearranging the equation gives:
\[ \alpha^2 \approx \frac{K_a}{c} \]
Taking the square root of both sides:
\[ \alpha \approx \sqrt{\frac{K_a}{c}} \]

### Step 7: Find the hydrogen ion concentration
The concentration of hydrogen ions \( [H^+] \) is given by:
\[ [H^+] = c\alpha \]
Substituting the expression for \( \alpha \):
\[ [H^+] \approx c \sqrt{\frac{K_a}{c}} \]
This simplifies to:
\[ [H^+] \approx \sqrt{c K_a} \]

### Conclusion
Thus, the hydrogen ion concentration in a weak acid of dissociation constant \( K_a \) and concentration \( c \) is nearly equal to:
\[ [H^+] \approx \sqrt{c K_a} \]