At what temperature is the root-mean-square speed of nitrogen molecules equal to the root-mean-square speed of hydrogen molecules at $20.0$°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 H₂ is twice the molar mass of hydrogen atoms, and similarly for N₂.)

Verified step by step guidance

1. First, understand the concept of root-mean-square speed (v_rms) for gas molecules, which is given by the formula: $\sqrt{\frac{3kT}{m}}$, where $k$ is the Boltzmann constant, $T$ is the temperature in Kelvin, and $m$ is the mass of a molecule.
2. Next, convert the given temperature of hydrogen molecules from Celsius to Kelvin using the formula: $T=T$_C+273.15. For hydrogen at 20.0°C, this becomes $293.15$ K.
3. Determine the molar mass of nitrogen (N2) and hydrogen (H2) using Appendix D. The molar mass of nitrogen atoms is approximately 14 g/mol, so for N2 it is $28$ g/mol. The molar mass of hydrogen atoms is approximately 1 g/mol, so for H2 it is $2$ g/mol.
4. Set up the equation for the root-mean-square speed equality: $\sqrt{\frac{3kT}{}}$_Nm_N = $\sqrt{\frac{3k293.15}{m}}$_H, where $T$_N is the temperature of nitrogen molecules, $m$_N is the mass of a nitrogen molecule, and $m$_H is the mass of a hydrogen molecule.
5. Solve for $T$_N by equating the expressions and simplifying: $\frac{T}{}$_N=m_Nm_H×293.15. Substitute the known values for $m$_N and $m$_H to find the temperature at which the root-mean-square speeds are equal.

Key Concepts

Root-Mean-Square Speed

The root-mean-square speed is a measure of the average speed of particles in a gas, derived from the kinetic theory of gases. It is calculated using the formula v_rms = sqrt(3kT/m), where k is the Boltzmann constant, T is the temperature in Kelvin, and m is the mass of a molecule. This concept helps compare the speeds of different gas molecules at varying temperatures.

Molar Mass

Molar mass is the mass of one mole of a substance, typically expressed in grams per mole (g/mol). It is crucial for calculating the mass of individual molecules, which is needed to determine their root-mean-square speed. For diatomic molecules like H2 and N2, the molar mass is twice that of the individual atoms, reflecting the combined mass of the two atoms in the molecule.

Temperature Conversion

Temperature conversion is essential for calculations involving the kinetic theory of gases, as temperatures must be in Kelvin. To convert from Celsius to Kelvin, add 273.15 to the Celsius temperature. This conversion ensures that the temperature is in the correct unit for use in the root-mean-square speed formula, allowing for accurate comparisons between different gases.