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Molar mass is a key concept in chemistry that helps to determine how much a substance weighs when measured in terms of its particles. It is expressed as grams per mole (g/mol), which simplifies the process of calculating the amounts of substances involved in chemical reactions. For example, the molar mass of water vapor, () is approximately 18 g/mol, while for carbon dioxide (CO2), it is about 44 g/mol.

This measurement lets us predict how much a particular mole of a compound would weigh. The molar mass is crucial when working with the root-mean-square (rms) speed of gas molecules. A gas with a larger molar mass will typically have a lower rms speed at the same temperature because heavier molecules move more slowly. This relationship forms the basis of calculating speeds of gases in physical chemistry problems.

In the exercise, comparing molar masses of two gases helps us understand their relative speeds. Since the molar mass of CO2 is larger than that of H2O, CO2 molecules will move slower than the water vapor molecules when they are at the same temperature.

The ideal gas constant (R) is an important constant used in the calculation of gas behavior, especially in the context of the ideal gas law and related equations. It allows us to link together various properties of gases such as pressure, volume, and temperature.

In the context of the rms speed calculations, the ideal gas constant, which is approximately 8.314 J/(mol·K), helps determine how fast molecules are moving based on their molar mass and the temperature. The equation connecting these elements is: \[ v_{rms} = \sqrt{\frac{3RT}{M}} \] where,

• \(v_{rms}\) is the root-mean-square speed,
• \(R\) is the ideal gas constant,
• \(T\) is the absolute temperature, and
• \(M\) is the molar mass.

With R as a known factor, solving for the rms speed becomes more straightforward once the temperature and molar mass are established. This makes the ideal gas constant an indispensable tool when calculating how fast different gas molecules will move under similar conditions.

Temperature is a measure of the average kinetic energy of the particles in a substance. In the context of gaseous molecules, it plays a pivotal role in determining how they move.

In our problem, both water vapor and carbon dioxide are at the same temperature, which simplifies the calculations by enabling direct comparisons between their rms speeds. The equation \( v_{rms} = \sqrt{\frac{3RT}{M}} \) demonstrates how temperature impacts the rms speed of molecules. As temperature increases, so does the kinetic energy of the molecules, resulting in higher rms speeds. Conversely, lower temperatures slow down molecular motion.

Since both gases are at the same temperature in the exercise, their rms speeds differ only because of their molar masses. If temperature were different, it would directly affect the speed calculations, indicating temperature's crucial role in determining molecular motion in gases.