To solve this, we need to use the Henderson-Hasselbalch equation, which is a useful equation for calculating the pH of a buffer solution. The equation is:

[ \text{pH} = \text{p}K_b + \log \left( \frac{[\text{base}]}{[\text{acid}]} \right) ]

Where:

• $\text{pH}$ is the desired pH of the buffer.
• $\text{p}K_b$ is the negative logarithm of the base dissociation constant, $K_b$, for CH₃NH₂.
• $[\text{base}]$ is the concentration of CH₃NH₂ (the weak base).
• $[\text{acid}]$ is the concentration of CH₃NH₃⁺ (the conjugate acid).

Step-by-Step Solution:

1. Calculate $\text{p}K_b$:

Given $K_b = 4.4 \times 10^{-4}$, we can calculate $\text{p}K_b$ using the formula:

$\text{p}K_b = -\log(K_b) = -\log(4.4 \times 10^{-4}) \approx 3.36$

2. Use the Henderson-Hasselbalch equation:

We know that: $\text{pH} = 9.70 \quad \text{and} \quad \text{p}K_b \approx 3.36$

Substitute the values into the Henderson-Hasselbalch equation:

[ 9.70 = 3.36 + \log \left( \frac{[\text{base}]}{[\text{acid}]} \right) ]

Rearrange the equation to solve for the ratio of base to acid:

[ 9.70 - 3.36 = \log \left( \frac{[\text{base}]}{[\text{acid}]} \right) ]

[ 6.34 = \log \left( \frac{[\text{base}]}{[\text{acid}]} \right) ]

Now, solve for the ratio:

[ \frac{[\text{base}]}{[\text{acid}]} = 10^{6.34} \approx 2.19 \times 10^{6} ]

Final Answer:

The ratio of CH₃NH₂ (base) to CH₃NH₃⁺ (acid) needed to prepare a pH 9.70 buffer is approximately 2.19 × 10⁶.

Would you like further details on this or clarification on any step? Here are a few related questions that might interest you:

1. What would happen to the pH of the buffer if the ratio of CH₃NH₂ to CH₃NH₃⁺ changes significantly?
2. How would the pH of the buffer change if a stronger acid or base was added?
3. Can you calculate the pH of a buffer if the concentrations of base and acid are known?
4. What is the relationship between $K_b$ and $K_a$ in weak acid-base equilibria?
5. How do you calculate the pH of a buffer solution when both the acid and the base are weak?

Tip: When dealing with buffers, remember that the pH is most sensitive when the concentrations of the acid and base are similar. If the ratio is very high or very low, the buffer capacity might be weaker.