Understanding the Problem
To determine the number of molecules in 32g of gas X, we first need to analyze the relationship between the diffusion rates of CH4 and gas X.
Diffusion Rate Relationship
- CH4 (methane) diffuses two times faster than gas X.
- According to Graham's Law of Effusion, the rate of diffusion is inversely proportional to the square root of the molar mass.
Applying Graham's Law
- Let the molar mass of CH4 be 16 g/mol.
- If CH4 diffuses two times faster, we can express this as:
Rate(CH4) / Rate(gas X) = √(Molar Mass(gas X) / Molar Mass(CH4))
- Given that Rate(CH4) = 2 * Rate(gas X), we can replace it in the equation:
2 = √(Molar Mass(gas X) / 16)
- Squaring both sides gives:
4 = Molar Mass(gas X) / 16
Molar Mass(gas X) = 64 g/mol.
Calculating the Number of Molecules
- Now that we have the molar mass of gas X, we can find the number of moles in 32g:
Moles of gas X = Mass / Molar Mass = 32g / 64 g/mol = 0.5 moles.
- To find the number of molecules, use Avogadro's number (6.022 x 10^23 molecules/mol):
Number of molecules = Moles x Avogadro's number = 0.5 moles x 6.022 x 10^23 molecules/mol = 3.011 x 10^23 molecules.
Conclusion
The number of molecules present in 32g of gas X is approximately 3.011 x 10^23 molecules. This analysis highlights the importance of understanding diffusion rates and the relationship between mass and molecular quantity.