Question

Gas A is composed of diatomic molecules (two atoms per molecule) of a pure element. Gas B is composed of triatomic molecules (three atoms per molecule) of another pure clement. A volume of gas \(B\) is found to be three times more massive than an cqual volume of gas A. How does the mass of an atom of gas B compare with the mass of an atom of gas A?

Short answer

The mass of an atom of gas B is twice that of an atom of gas A.

Step-by-step solution

Define Variables

Let the mass of a single atom of gas A be \( M_A \) and the mass of a single atom of gas B be \( M_B \). The mass of a diatomic molecule of gas A is thus \( 2M_A \), while the mass of a triatomic molecule of gas B is \( 3M_B \).

Express Given Condition Mathematically

We know that the mass of a volume \( V \) of gas B is three times the mass of the same volume of gas A. Therefore, if \( n \) is the number of molecules in volume \( V \) for each gas, then for gas A: \( n \cdot 2M_A \) and for gas B: \( n \cdot 3M_B \). The equation representing this is: \( n \cdot 3M_B = 3(n \cdot 2M_A) \).

Simplify the Equation

Simplify the equation from Step 2: \( 3nM_B = 6nM_A \). Now observe that \( n \) cancels out from both sides, leaving us with \( 3M_B = 6M_A \). Divide both sides by 3 to get \( M_B = 2M_A \).

Interpret the Result

The simplification \( M_B = 2M_A \) indicates that the mass of an atom of gas B is twice the mass of an atom of gas A.

Key concepts

Diatomic Molecules

Diatomic molecules are composed of two atoms that are chemically bonded together. They can consist of two identical atoms (homonuclear), such as in oxygen (\(O_2\)), or two different atoms (heteronuclear), such as in carbon monoxide (\(CO\)). For gases composed of diatomic molecules, understanding the molecule's structure is fundamental to predicting its behavior and properties.

In the analysis of gas A in the provided problem, which comprises diatomic molecules of a pure element, the molecular weight will be the sum of the masses of the two atoms forming each molecule.

• Diatomic Elements: Common examples include hydrogen (\(H_2\)), nitrogen (\(N_2\)), and chlorine (\(Cl_2\)).
• Properties: Diatomic gases typically share certain properties, such as similar boiling points and bond energies.

Triatomic Molecules

Triatomic molecules consist of three atoms bonded together. These molecules may form a linear or bent shape structure, influencing their chemical properties and interactions.

For gas B in the problem, which is composed of triatomic molecules, this structure impacts its total molecular weight, as it is calculated based on the collective mass of all three atoms.

• Common Triatomic Molecules: Ozone (\(O_3\)) and carbon dioxide (\(CO_2\)) are popular examples found in nature.
• Effects of Structure: The shape and bond angles of triatomic molecules, such as in water (\(H_2O\)) which has a bent shape, are significant in determining the molecule's unique physical and chemical properties.

Atomic Mass Comparison

The comparison of atomic masses is crucial to understanding how different gases interact under the same conditions. In the problem, careful attention is paid to comparing the masses of individual atoms of gases A and B.

To find the relationship between the masses of the atoms making up gases A and B, we leverage their molecular structures. For each diatomic molecule of gas A, we calculate its mass as twice the atomic mass of one constituent atom (\(2M_A\)), while for triatomic gas B, the mass sums up to thrice its atomic component (\(3M_B\)).

• Mathematical Resolution: By setting the total mass of gas B as three times that of gas A per equivalent volume, the solution simplifies to the atomic mass of gas B being twice that of gas A (\(M_B = 2M_A\)).
• Result Interpretation: Such comparisons reveal not only ratios but also help in calculating practical implications in thermodynamics and stoichiometry.