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The root-mean-square speed (\(v_{rms}\)) is a measure of the speed of particles in a gas. It provides an idea of how fast molecules are moving within a given sample.
The concept of root-mean-square speed is vital in the kinetic theory of gases, which explains the behavior of gases in terms of particle motion. To calculate it, we use the formula:\[v_{rms} = \sqrt{\frac{3RT}{M}}\]where \(R\) is the gas constant, \(T\) is the temperature in Kelvin, and \(M\) is the molar mass of the gas.
This formula highlights that the speed of gas molecules is dependent on both the temperature and the mass. Lighter molecules and higher temperatures lead to higher speeds. This is because lighter molecules require less energy to increase their speed. Similarly, higher temperatures suggest more kinetic energy available to the molecules.

Molar mass is the mass of a given substance (chemical element or chemical compound) divided by the amount of substance. It is usually expressed in units of \(g/mol\).
In the context of gases, knowing the molar mass is crucial for computing the root-mean-square speed. When comparing two gases, as in the exercise with nitrogen (\(N_2\)) and hydrogen (\(H_2\)), the molar mass helps determine at what conditions similar speeds are achieved. For instance, the molar mass of hydrogen gas (\(H_2\)) is approximately 2 \(g/mol\), as hydrogen has an atomic mass of 1 for each atom, and there are two atoms in a hydrogen molecule.
Similarly, nitrogen molecules (\(N_2\)) have a molar mass of around 28 \(g/mol\) because nitrogen has an atomic mass of 14, and the nitrogen molecule consists of two atoms.
Understanding the molar mass is essential for calculating how temperature variations affect the speeds of different gases, allowing for the comparative analysis of molecular motion.

Temperature conversion is an important step when dealing with gas laws and kinetic molecular theory. Different scientific contexts require temperatures in different units, typically Celsius or Kelvin.
For any kinetic calculations involving gases, temperature must be in Kelvin. This is because Kelvin is an absolute scale where 0 K represents absolute zero—the point at which molecular motion theoretically stops.
The conversion from Celsius (\(^{\circ}C\)) to Kelvin (\(K\)) is simple:

• Add 273.15 to the Celsius temperature.

For the given problem, the hydrogen molecules are initially at 20°C, which is converted to:\[20 + 273.15 = 293.15\,\]approximated as 293 K for ease of calculation.
Accurate temperature conversion ensures that computations of gas properties, like the root-mean-square speed, are based on consistent and correct units.

Molecular mass refers to the sum of the atomic masses of all atoms in a molecule. Unlike molar mass, which is expressed in grams per mole, molecular mass is measured in atomic mass units (amu).
To find the molecular mass of a gas, one simply adds up the atomic masses of all the atoms in the molecule, as listed in the periodic table. For instance, in a nitrogen molecule (\(N_2\)), each nitrogen atom has an atomic mass of approximately 14 amu, resulting in a total molecular mass of about 28 amu for the molecule.
Similarly, a hydrogen molecule (\(H_2\)) has 2 atoms of hydrogen, each with an atomic mass of about 1 amu, totaling approximately 2 amu for the molecule. Although molecular mass and molar mass are related, it is important to distinguish between the two.

• Molecular mass gives an idea of the individual molecule's mass.
• Molar mass provides information on the mass of a mole of these molecules and assists in converting between microscopic and macroscopic quantities.

Understanding molecular mass is fundamental when working with equations and concepts in gas theory, ensuring clear and accurate molecular characterizations.