Calculating the Number of Molecules in 0.8g of Methane (CH4)

Methane (CH4) is one of the most abundant gases in the Earth's atmosphere, used widely in various applications, including energy generation and as a fuel source. Often, we need to determine the number of molecules within a given mass of methane, a process commonly required in both educational and professional contexts. This article will guide you through the steps to calculate the number of molecules in 0.8 grams of methane using fundamental chemistry principles.

Determining the Molar Mass of Methane (CH4)

The molecular formula for methane (CH4) is made up of one carbon atom and four hydrogen atoms. The atomic masses for these elements are:

Carbon (C): 12.01 g/mol Hydrogen (H): 1.008 g/mol

To find the molar mass of methane, we use the formula:

Molecular mass of CH4 Atomic mass of C 4 × Atomic mass of H

Substituting the atomic masses:

Molecular mass of CH4 12.01 g/mol 4 × 1.008 g/mol 12.01 g/mol 4.032 g/mol 16.042 g/mol

Calculating the Number of Moles in 0.8g of Methane

Now that we have the molar mass of methane, we can use the formula:

Number of moles (n) Mass (m) / Molar mass (Mw)

Substituting the values:

Number of moles (n) 0.8 g / 16.042 g/mol ≈ 0.0498 mol

Calculating the Number of Molecules Using Avogadro's Number

Avogadro's number (NA) is approximately 6.022 × 1023 molecules/mol. We can use this constant along with the number of moles to find the number of molecules:

Number of molecules Number of moles × Avogadro's number

Substituting the values:

Number of molecules ≈ 0.0498 mol × 6.022 × 1023 molecules/mol 3.00 × 1022 molecules

Verification and Summary

Three alternative methods to demonstrate the calculation have been provided, reinforcing our initial calculation. Each method uses the formula ( n frac{m}{Mw} ) to find the number of moles, then multiplies by Avogadro's number to find the number of molecules. Here are the calculations for your reference:

Method 1: Simple Calculation

Number of moles: ( n frac{0.8 text{ g}}{16.043 text{ g/mol}} approx 0.0499 text{ mol} )

Number of molecules: ( 0.0499 text{ mol} times 6.022 times 10^{23} text{ molecules/mol} approx 3.005 times 10^{23} text{ molecules} )

Method 2: Using 0.05 Mole Approximation

Mass of methane: 0.8 g

Molar mass of methane: 16 g/mol

Number of moles: ( n frac{0.8 text{ g}}{16 text{ g/mol}} 0.05 text{ mol} )

Number of molecules: ( 0.05 text{ mol} times 6.022 times 10^{23} text{ molecules/mol} 3 times 10^{22} text{ molecules} )

Method 3: Applying Avogadro's Law and Ideal Gas Law

Number of moles: ( n frac{0.8 text{ g}}{16.043 text{ g/mol}} approx 0.0499 text{ mol} )

Avogadro's constant: ( N_A 6.022 times 10^{23} text{ molecules/mol} )

Number of molecules: ( N n times N_A 0.0499 text{ mol} times 6.022 times 10^{23} text{ molecules/mol} approx 3.005 times 10^{23} text{ molecules} )

Both the primary calculation and the alternative methods yield similar results, confirming the accuracy of our calculations.

Conclusion

Understanding how to calculate the number of molecules in a given mass of methane is crucial for a variety of applications, from chemical reactions and thermodynamics to atmospheric studies and industrial processes. The formula ( n frac{m}{Mw} ) and the application of Avogadro's number provide a reliable and straightforward method for this calculation. By following these steps, you can ensure accuracy in your molecular calculations, enhancing your understanding of methane and its applications.