240cc of `SO_2` gas diffused through a porous membrane in 20 minutes. Under similar conditions, 720 cc of another gas diffused in 30 minutes. Find the molecular mass of the gas.

Answer

To solve the problem, we will use Graham's law of effusion, which states that the rate of diffusion of a gas is inversely proportional to the square root of its molar mass (molecular weight).

### Step-by-Step Solution:

1. **Identify the Given Data:**
- Volume of `SO2` = 240 cc
- Time for `SO2` = 20 minutes
- Volume of unknown gas = 720 cc
- Time for unknown gas = 30 minutes

2. **Calculate the Rate of Diffusion:**
- The rate of diffusion is given by the formula:
\[ \text{Rate} = \frac{\text{Volume}}{\text{Time}} \]
- For `SO2`:
\[ \text{Rate}_{SO2} = \frac{240 \text{ cc}}{20 \text{ min}} = 12 \text{ cc/min} \]
- For the unknown gas:
\[ \text{Rate}_{\text{unknown}} = \frac{720 \text{ cc}}{30 \text{ min}} = 24 \text{ cc/min} \]

3. **Apply Graham's Law:**
- According to Graham's law:
\[ \frac{\text{Rate}_{SO2}}{\text{Rate}_{\text{unknown}}} = \sqrt{\frac{M_{\text{unknown}}}{M_{SO2}}} \]
- Where \( M \) is the molecular mass.
- Rearranging gives:
\[ \frac{12}{24} = \sqrt{\frac{M_{\text{unknown}}}{M_{SO2}}} \]
- Simplifying the left side:
\[ \frac{1}{2} = \sqrt{\frac{M_{\text{unknown}}}{M_{SO2}}} \]

4. **Square Both Sides:**
- Squaring both sides to eliminate the square root:
\[ \left(\frac{1}{2}\right)^2 = \frac{M_{\text{unknown}}}{M_{SO2}} \]
- This simplifies to:
\[ \frac{1}{4} = \frac{M_{\text{unknown}}}{M_{SO2}} \]

5. **Find the Molecular Mass of `SO2`:**
- The molecular mass of `SO2` is calculated as follows:
- Atomic mass of Sulfur (S) = 32 g/mol
- Atomic mass of Oxygen (O) = 16 g/mol
- Therefore, for `SO2`:
\[ M_{SO2} = 32 + (16 \times 2) = 32 + 32 = 64 \text{ g/mol} \]

6. **Substitute and Solve for \( M_{\text{unknown}} \):**
- Substitute \( M_{SO2} \) into the equation:
\[ \frac{1}{4} = \frac{M_{\text{unknown}}}{64} \]
- Rearranging gives:
\[ M_{\text{unknown}} = 64 \times \frac{1}{4} = 16 \text{ g/mol} \]

### Conclusion:
The molecular mass of the unknown gas is **16 g/mol**.