compute gram molecular weight of H2 gas if 4 gram of H2 gas contains 24.09 * 10^23 atoms?

To find the gram molecular weight of hydrogen gas (H2), we first need to understand a few key concepts, including the relationship between moles, atoms, and molecular weight. Let's break this down step by step.

Understanding the Basics

Hydrogen gas (H2) consists of two hydrogen atoms bonded together. The molecular weight of a substance is the mass of one mole of that substance, measured in grams per mole (g/mol). For hydrogen, the atomic weight of a single hydrogen atom is approximately 1 gram per mole. Therefore, the molecular weight of H2 is about 2 grams per mole (since there are two hydrogen atoms in each molecule).

Calculating the Number of Moles

We know that 4 grams of H2 gas contains 24.09 x 10^23 atoms. To find the number of moles of H2 in 4 grams, we can use Avogadro's number, which is approximately 6.022 x 10^23 atoms per mole. The formula to calculate the number of moles (n) is:

• n = Number of atoms / Avogadro's number

Substituting the values we have:

• n = (24.09 x 10^23 atoms) / (6.022 x 10^23 atoms/mole)

Calculating this gives:

• n ≈ 4 moles of H2

Finding the Gram Molecular Weight

Now that we know we have 4 moles of H2 in 4 grams, we can calculate the gram molecular weight. The gram molecular weight is defined as the mass of one mole of a substance. Since we have 4 grams of H2 gas, we can find the molecular weight by dividing the mass by the number of moles:

• Molecular weight = Mass / Moles

Substituting the values:

• Molecular weight = 4 grams / 4 moles

This simplifies to:

• Molecular weight = 1 gram/mole

Final Thoughts

However, this result seems inconsistent with our earlier understanding that the molecular weight of H2 should be around 2 g/mol. The discrepancy arises because we calculated the number of moles based on the number of atoms rather than the number of molecules. Since each molecule of H2 contains 2 atoms, we should actually divide the total number of atoms by 2 to find the number of H2 molecules:

• Number of H2 molecules = 24.09 x 10^23 atoms / 2 = 12.045 x 10^23 molecules

Now, using Avogadro's number:

• n = (12.045 x 10^23 molecules) / (6.022 x 10^23 molecules/mole) ≈ 2 moles of H2

Finally, we can recalculate the gram molecular weight:

• Molecular weight = 4 grams / 2 moles = 2 grams/mole

This aligns perfectly with the known molecular weight of hydrogen gas. So, the gram molecular weight of H2 gas is indeed 2 grams per mole.