How many grams of `H_(2)SO_(4)` is/are to be dissolved to prepare 200 mL aqueous solution having concentration of `[H_(3)O^(+)]` ions is 1 M at `25^(@)C` temperature?

Options

A. 19.6 g
B. 0.98 g
C. 4.9 g
D. 9.8 g

Explanation

To determine how many grams of \( H_2SO_4 \) are needed to prepare a 200 mL aqueous solution with a concentration of \( [H_3O^+] \) ions equal to 1 M at \( 25^\circ C \), we can follow these steps:

### Step 1: Understand the relationship between \( H_2SO_4 \) and \( H_3O^+ \)
Sulfuric acid (\( H_2SO_4 \)) is a strong acid that dissociates in water to produce \( H_3O^+ \) ions. The dissociation can be represented as:
\[ H_2SO_4 \rightarrow 2 H_3O^+ + SO_4^{2-} \]
This means that 1 mole of \( H_2SO_4 \) produces 2 moles of \( H_3O^+ \).

### Step 2: Calculate the moles of \( H_3O^+ \) needed
Since we want a concentration of 1 M in a 200 mL solution, we can calculate the number of moles of \( H_3O^+ \) required:
\[ \text{Moles of } H_3O^+ = \text{Concentration} \times \text{Volume (in L)} = 1 \, \text{mol/L} \times 0.200 \, \text{L} = 0.200 \, \text{mol} \]

### Step 3: Relate moles of \( H_3O^+ \) to moles of \( H_2SO_4 \)
Since 1 mole of \( H_2SO_4 \) produces 2 moles of \( H_3O^+ \), we can find the moles of \( H_2SO_4 \) required:
\[ \text{Moles of } H_2SO_4 = \frac{\text{Moles of } H_3O^+}{2} = \frac{0.200 \, \text{mol}}{2} = 0.100 \, \text{mol} \]

### Step 4: Calculate the mass of \( H_2SO_4 \)
Next, we need to calculate the mass of \( H_2SO_4 \) using its molar mass. The molar mass of \( H_2SO_4 \) is calculated as follows:
- Hydrogen (H): 1 g/mol × 2 = 2 g/mol
- Sulfur (S): 32 g/mol × 1 = 32 g/mol
- Oxygen (O): 16 g/mol × 4 = 64 g/mol

Total molar mass of \( H_2SO_4 \):
\[ \text{Molar mass of } H_2SO_4 = 2 + 32 + 64 = 98 \, \text{g/mol} \]

Now, we can calculate the mass of \( H_2SO_4 \) needed:
\[ \text{Mass of } H_2SO_4 = \text{Moles} \times \text{Molar Mass} = 0.100 \, \text{mol} \times 98 \, \text{g/mol} = 9.8 \, \text{g} \]

### Final Answer
To prepare a 200 mL solution with a concentration of \( [H_3O^+] \) ions equal to 1 M, you need to dissolve **9.8 grams of \( H_2SO_4 \)**.

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