Question

Select the correct statement(s) regarding buffer solution: $CH_3COOH(X \text{ molar}) + CH_3COONa(Y \text{ molar})$

• A. At constant buffer concentration, buffer capacity increases to its maximum value if $\frac{Y}{X}$ approaches 1.
• B. At constant buffer concentration, variation of index of buffer with Y is parabolic in nature.
• C. Keeping $\frac{Y}{X}=1, \beta$ (buffer capacity) vs Y can be represented by
• D. On dilution, pH of the buffer solution remains constant whereas its buffer capacity increases.

Answer 1

Answer: (A), (B), (C)

(1), (2), (3)

Answer 2

The buffer solution consists of a weak acid ($CH_3COOH$, concentration $X$) and its conjugate base ($CH_3COO^-$, from $CH_3COONa$, concentration $Y$).

Statement (1): At constant buffer concentration, buffer capacity increases to its maximum value if $\frac{Y}{X}$ approaches 1.

The buffer capacity ($\beta$) for a weak acid-conjugate base buffer is given by $\beta = 2.303 \frac{[Acid][Salt]}{[Acid] + [Salt]}$. Let the total buffer concentration be $C = [Acid] + [Salt] = X + Y$. Then $\beta = 2.303 \frac{X Y}{X+Y} = 2.303 \frac{X Y}{C}$. We want to maximize $\beta$ for a constant $C$. We know that for a fixed sum $X+Y=C$, the product $XY$ is maximum when $X=Y$. If $X=Y$, then $\frac{Y}{X}=1$. So, at constant buffer concentration, the buffer capacity is maximum when the ratio $\frac{Y}{X} = 1$. As the ratio $\frac{Y}{X}$ approaches 1, the buffer capacity increases towards its maximum value. This statement is correct.

Statement (2): At constant buffer concentration, variation of index of buffer with Y is parabolic in nature.

Let the constant buffer concentration be $C = X+Y$. So $X = C-Y$. The buffer capacity is $\beta = 2.303 \frac{X Y}{X+Y} = 2.303 \frac{(C-Y) Y}{C} = \frac{2.303}{C} (CY - Y^2)$. This is a quadratic function of $Y$ of the form $\beta = aY^2 + bY$ where $a = -\frac{2.303}{C}$ and $b = 2.303$. The graph of a quadratic function is a parabola. Since the coefficient of $Y^2$ is negative, the parabola opens downwards. The variable $Y$ represents concentration, so $Y \ge 0$. Also, $X = C-Y \ge 0$, so $Y \le C$. Thus, the domain of $Y$ is $0 \le Y \le C$. The variation of buffer capacity with $Y$ is parabolic in nature over this range. This statement is correct.

Statement (3): Keeping $\frac{Y}{X}=1$, $\beta$ (buffer capacity) vs Y can be represented by the given figure.

If $\frac{Y}{X} = 1$, then $Y=X$. The buffer capacity is $\beta = 2.303 \frac{X Y}{X+Y}$. Substituting $X=Y$, we get $\beta = 2.303 \frac{Y \cdot Y}{Y+Y} = 2.303 \frac{Y^2}{2Y} = 2.303 \frac{Y}{2}$. So, $\beta = \left(\frac{2.303}{2}\right) Y$. This is a linear relationship between $\beta$ and $Y$ with a positive slope of $\frac{2.303}{2}$ and a y-intercept of 0. The given figure shows a straight line passing through the origin with a positive slope, which represents a linear relationship between $\beta$ and $Y$ where $\beta$ is directly proportional to $Y$. This is consistent with the derived equation. This statement is correct.

Statement (4): On dilution, pH of the buffer solution remains constant whereas its buffer capacity increases.

According to the Henderson-Hasselbalch equation, $pH = pK_a + \log \frac{[Salt]}{[Acid]} = pK_a + \log \frac{Y}{X}$. When a buffer solution is diluted by a factor $D$, the new concentrations are $X' = X/D$ and $Y' = Y/D$. The new pH is $pH' = pK_a + \log \frac{Y'}{X'} = pK_a + \log \frac{Y/D}{X/D} = pK_a + \log \frac{Y}{X} = pH$. So, the pH of the buffer solution remains constant on dilution (assuming concentrations are not extremely low). The original buffer capacity is $\beta = 2.303 \frac{X Y}{X+Y}$. After dilution, the new buffer capacity is $\beta' = 2.303 \frac{X' Y'}{X'+Y'} = 2.303 \frac{(X/D)(Y/D)}{(X/D)+(Y/D)} = 2.303 \frac{XY/D^2}{(X+Y)/D} = 2.303 \frac{XY}{D(X+Y)} = \frac{1}{D} \left( 2.303 \frac{XY}{X+Y} \right) = \frac{\beta}{D}$. Since dilution means $D>1$, the new buffer capacity $\beta'$ is less than the original buffer capacity $\beta$. So, on dilution, the buffer capacity decreases, not increases. This statement is incorrect.

Based on the analysis, statements (1), (2), and (3) are correct.