In what mass of water must 25g of `CuSO_(4)*5H_(2)O` be dissolved to obtain an `8%` solution of `CuSO_(4)` ?

Answer

To solve the problem of how much mass of water is needed to dissolve 25 g of `CuSO4·5H2O` to obtain an 8% solution of `CuSO4`, we can follow these steps:

### Step 1: Calculate the molar mass of `CuSO4·5H2O`
The molar mass of `CuSO4·5H2O` is calculated as follows:
- Molar mass of `Cu` = 63.55 g/mol
- Molar mass of `S` = 32.07 g/mol
- Molar mass of `O` = 16.00 g/mol (4 O in `CuSO4`)
- Molar mass of `H2O` = 18.02 g/mol (5 H2O)

Calculating:
- Molar mass of `CuSO4` = 63.55 + 32.07 + (4 × 16.00) = 159.62 g/mol
- Molar mass of `5H2O` = 5 × 18.02 = 90.10 g/mol
- Total molar mass of `CuSO4·5H2O` = 159.62 + 90.10 = 249.72 g/mol

### Step 2: Calculate the mass of `CuSO4` in 25 g of `CuSO4·5H2O`
To find out how much `CuSO4` is present in 25 g of `CuSO4·5H2O`, we use the ratio of the molar mass of `CuSO4` to the molar mass of `CuSO4·5H2O`.

\[ \text{Mass of } CuSO4 = \left( \frac{159.62 \text{ g/mol}}{249.72 \text{ g/mol}} \right) \times 25 \text{ g} \]

Calculating:
\[ \text{Mass of } CuSO4 = \left( \frac{159.62}{249.72} \right) \times 25 \approx 15.98 \text{ g} \]

### Step 3: Set up the equation for the 8% solution
An 8% solution means that 8 g of solute (in this case, `CuSO4`) is present in 100 g of solution. Let \( X \) be the total mass of the solution.

\[ \frac{15.98 \text{ g}}{X} = 0.08 \]

### Step 4: Solve for \( X \)
Rearranging the equation gives us:

\[ X = \frac{15.98 \text{ g}}{0.08} = 199.75 \text{ g} \]

### Step 5: Calculate the mass of water needed
The mass of water can be found by subtracting the mass of `CuSO4·5H2O` from the total mass of the solution.

\[ \text{Mass of water} = X - \text{mass of } CuSO4·5H2O \]
\[ \text{Mass of water} = 199.75 \text{ g} - 25 \text{ g} = 174.75 \text{ g} \]

### Final Answer
The mass of water needed is **174.75 g**.

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