Nalorphene `(C_(19)H_(22)NO_(3))`, similar to morphine , is used to combat withdrawal symptoms in narcotic users. The dose of nalorphene generally given is `1.5 mg`. Calculate the mass of solution of `1.5xx10^(-3)m` aqueous solution required for the above dose.

Explanation

To solve the problem, we need to calculate the mass of a solution that has a molality of \(1.5 \times 10^{-3} \, \text{mol/kg}\) and contains a dose of \(1.5 \, \text{mg}\) of nalorphene \((C_{19}H_{22}NO_{3})\).

### Step-by-Step Solution:

1. **Convert the Dose from mg to g**:
\[ \text{Dose} = 1.5 \, \text{mg} = 1.5 \times 10^{-3} \, \text{g} \]

2. **Calculate the Molar Mass of Nalorphene**:
- The molecular formula is \(C_{19}H_{22}NO_{3}\).
- Calculate the molar mass:
\[ \text{Molar mass} = (19 \times 12) + (22 \times 1) + (14) + (3 \times 16) \]
\[ = 228 + 22 + 14 + 48 = 312 \, \text{g/mol} \]

3. **Calculate the Number of Moles of Nalorphene**:
\[ \text{Moles of Nalorphene} = \frac{\text{mass}}{\text{molar mass}} = \frac{1.5 \times 10^{-3} \, \text{g}}{312 \, \text{g/mol}} \approx 4.8 \times 10^{-6} \, \text{mol} \]

4. **Use the Definition of Molality**:
- Molality (\(m\)) is defined as:
\[ m = \frac{\text{moles of solute}}{\text{mass of solvent (kg)}} \]
- Rearranging gives us:
\[ \text{mass of solvent (kg)} = \frac{\text{moles of solute}}{m} \]
- Substitute the values:
\[ \text{mass of solvent (kg)} = \frac{4.8 \times 10^{-6} \, \text{mol}}{1.5 \times 10^{-3} \, \text{mol/kg}} \approx 0.0032 \, \text{kg} \]

5. **Convert Mass of Solvent to grams**:
\[ \text{mass of solvent (g)} = 0.0032 \, \text{kg} \times 1000 \, \text{g/kg} = 3.2 \, \text{g} \]

6. **Calculate the Mass of the Solution**:
- The mass of the solution is the sum of the mass of the solute and the mass of the solvent:
\[ \text{mass of solution} = \text{mass of solute} + \text{mass of solvent} \]
- Substitute the values:
\[ \text{mass of solution} = 1.5 \times 10^{-3} \, \text{g} + 3.2 \, \text{g} \approx 3.2015 \, \text{g} \]

### Final Answer:
The mass of the solution required for the dose of nalorphene is approximately **3.2015 g**.