To solve the problem, we will follow these steps:
### Step 1: Determine the initial concentration of H⁺ ions
We know that:
- The number of moles of H⁺ = \(2 \times 10^6\) mol
- The volume of the solution = 0.05 mL
First, we need to convert the volume from milliliters to liters:
\[
\text{Volume in liters} = 0.05 \, \text{mL} \times \frac{1 \, \text{L}}{1000 \, \text{mL}} = 0.00005 \, \text{L}
\]
Now, we can calculate the concentration of H⁺ ions:
\[
\text{Concentration} = \frac{\text{Number of moles}}{\text{Volume in liters}} = \frac{2 \times 10^6 \, \text{mol}}{0.00005 \, \text{L}} = 4 \times 10^{10} \, \text{mol L}^{-1}
\]
### Step 2: Identify the order of the reaction
The rate constant \(k\) is given as \(10^7 \, \text{mol L}^{-1} \, \text{s}^{-1}\). Since the unit of the rate constant is \( \text{mol L}^{-1} \, \text{s}^{-1}\), this indicates that the reaction is a zero-order reaction.
### Step 3: Use the zero-order reaction formula
For a zero-order reaction, the rate of disappearance can be expressed as:
\[
\text{Rate} = k = \frac{\Delta [A]}{\Delta t}
\]
Where \(\Delta [A]\) is the change in concentration and \(\Delta t\) is the time taken.
### Step 4: Calculate the time for disappearance of H⁺ ions
We can rearrange the formula to find the time:
\[
\Delta t = \frac{\Delta [A]}{k}
\]
Here, \(\Delta [A]\) is equal to the initial concentration since we are considering the complete disappearance of H⁺:
\[
\Delta [A] = 4 \times 10^{10} \, \text{mol L}^{-1}
\]
Substituting the values:
\[
\Delta t = \frac{4 \times 10^{10} \, \text{mol L}^{-1}}{10^7 \, \text{mol L}^{-1} \, \text{s}^{-1}} = 4 \times 10^3 \, \text{s}
\]
### Final Answer
The time required for the disappearance of H⁺ ions is \(4 \times 10^3\) seconds.
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