At what temperature is the root-mean-square speed of...

At what temperature is the root-mean-square speed of nitrogen molecules equal to the root-mean-square speed of hydrogen molecules at 20.0$^\circ$C? ($Hint$: Appendix D shows the molar mass (in g/mol) of each element under the chemical symbol for that element. The molar mass of H$_2$ is twice the molar mass of hydrogen atoms, and similarly for N$_2$.)

Key Concepts

Molar Mass

Molar mass is the mass of one mole of a substance and plays a crucial role in determining the molecular speed in gases. In the rms speed equation, the molar mass appears in the denominator, indicating that, for a given temperature, lighter molecules will have higher speeds compared to heavier molecules. Understanding molar mass is thus vital when analyzing and comparing the motion of different gases under the same conditions.

Temperature Conversion

Temperature must be expressed in an absolute scale, such as Kelvin, when used in gas law equations, including the rms speed equation. This conversion is essential because the rms speed formula is derived based on the absolute temperature, ensuring that computations correctly reflect the kinetic energy of the molecules. Converting temperatures from Celsius to Kelvin is a fundamental step in many thermodynamic calculations.

Kinetic Molecular Theory

This theory describes the behavior of gases in terms of their molecular motion. It links the temperature of a gas to the average kinetic energy of its molecules, explaining that higher temperatures correspond to higher molecular speeds. This connection is foundational for understanding phenomena such as pressure exerted by gases and the distribution of molecular speeds in a sample.

Root-Mean-Square Speed

The root-mean-square (rms) speed is a measure of the average speed of gas molecules, derived from the kinetic theory equation. It is defined mathematically as the square root of the average of the squares of the individual molecular speeds. The rms speed is related to the temperature and the molecular mass by the equation vrms = ?(3RT/M), making it a critical concept when comparing the speeds of different gases at various temperatures.

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