The sum of pH and `pK_(b)` for a basic buffer solution is 13. The ratio of the concentration of the base to that of the salt is

Options

A. 10
B. 1
C. 0.05
D. 0.1

Answer

To solve the problem step by step, we will use the information provided about the basic buffer solution and the relationship between pH, pK_b, and the concentrations of the base and salt.

### Step 1: Understanding the relationship
For a basic buffer solution, we know that:
\[ \text{pH} + \text{pK}_b = 14 \]

### Step 2: Given information
We are given that:
\[ \text{pH} + \text{pK}_b = 13 \]

### Step 3: Substitute the equation
From the first equation, we can express pK_b in terms of pH:
\[ \text{pK}_b = 14 - \text{pH} \]

### Step 4: Substitute into the given equation
Now, substituting this into the second equation:
\[ \text{pH} + (14 - \text{pH}) = 13 \]

### Step 5: Simplify the equation
This simplifies to:
\[ 14 = 13 \]
This means that we need to find the relationship between pH and pK_b directly.

### Step 6: Rearranging the equation
From the equation \( \text{pH} + \text{pK}_b = 13 \), we can express pK_b:
\[ \text{pK}_b = 13 - \text{pH} \]

### Step 7: Using the Henderson-Hasselbalch equation
The Henderson-Hasselbalch equation for a basic buffer is given by:
\[ \text{pH} = \text{pK}_b + \log\left(\frac{[\text{Salt}]}{[\text{Base}]}\right) \]

### Step 8: Substitute pK_b into the equation
Substituting \( \text{pK}_b = 13 - \text{pH} \) into the Henderson-Hasselbalch equation:
\[ \text{pH} = (13 - \text{pH}) + \log\left(\frac{[\text{Salt}]}{[\text{Base}]}\right) \]

### Step 9: Rearranging the equation
Rearranging gives:
\[ 2\text{pH} - 13 = \log\left(\frac{[\text{Salt}]}{[\text{Base}]}\right) \]

### Step 10: Solving for the ratio
Let’s assume \( \text{pH} = x \):
\[ 2x - 13 = \log\left(\frac{[\text{Salt}]}{[\text{Base}]}\right) \]

### Step 11: Exponentiating to find the ratio
Exponentiating both sides gives:
\[ 10^{(2x - 13)} = \frac{[\text{Salt}]}{[\text{Base}]} \]

### Step 12: Finding the ratio of base to salt
To find the ratio of the concentration of the base to that of the salt, we take the reciprocal:
\[ \frac{[\text{Base}]}{[\text{Salt}]} = 10^{(13 - 2x)} \]

### Step 13: Using the condition \( \text{pH} + \text{pK}_b = 13 \)
Since we know \( \text{pH} + \text{pK}_b = 13 \), we can substitute \( \text{pH} = 13 - \text{pK}_b \) into the equation to find the specific ratio.

### Final Calculation
If we assume \( \text{pH} = 7 \) (for example), then \( \text{pK}_b = 6 \):
\[ \frac{[\text{Base}]}{[\text{Salt}]} = 10^{(13 - 14)} = 10^{-1} = 0.1 \]

Thus, the ratio of the concentration of the base to that of the salt is:
\[ \frac{[\text{Base}]}{[\text{Salt}]} = 0.1 \]