At what temp do hydrogen molecules, H2, have the same rms speed as nitrogen molecules, N2, at 455 C? At what temperature do hydrogen molecules have the same average kinetic energy?
To find the temperature at which hydrogen molecules (H2) have the same root mean square (rms) speed as nitrogen molecules (N2) at 455 °C, we need to compare their kinetic energies using the kinetic theory of gases.
The root mean square speed (vrms) of a gas molecule can be calculated using the following formula:
vrms = √(3kT / m)
Where:
- vrms is the root mean square speed of the molecule
- k is the Boltzmann constant (1.38 × 10^-23 J/K)
- T is the temperature in Kelvin (K)
- m is the molar mass of the gas molecule
The molar mass of a hydrogen molecule (H2) is 2 g/mol, while the molar mass of a nitrogen molecule (N2) is 28 g/mol.
Step 1: Convert the given temperature to Kelvin
455 °C + 273.15 = 728.15 K
Step 2: Calculate the vrms for nitrogen (N2) at the given temperature
vrms(N2) = √(3kT / m(N2))
= √[(3 × 1.38 × 10^-23 J/K × 728.15 K) / 0.028 kg/mol]
≈ 515.4 m/s
Step 3: Calculate the temperature at which hydrogen (H2) exhibits the same vrms as nitrogen (N2)
vrms(H2) = vrms(N2)
√(3kT / m(H2)) = vrms(N2)
Squaring both sides:
3kT / m(H2) = vrms(N2)^2
Solving for T:
T = (vrms(N2)^2 × m(H2)) / (3k)
= (515.4 m/s)^2 × 0.002 kg/mol / (3 × 1.38 × 10^-23 J/K)
≈ 635 K
Therefore, hydrogen molecules (H2) at approximately 635 K (361 °C) will have the same rms speed as nitrogen molecules (N2) at 455 °C.
To find the temperature at which hydrogen molecules have the same average kinetic energy as nitrogen molecules, we can use the equation:
KEavg = (3/2)kT
Step 4: Calculate the average kinetic energy (KEavg) of nitrogen (N2) at 455 °C
KEavg(N2) = (3/2)kT
= (3/2) × (1.38 × 10^-23 J/K) × 728.15 K
≈ 2.81 × 10^-21 J
Step 5: Calculate the temperature at which hydrogen (H2) exhibits the same KEavg as nitrogen (N2)
KEavg(H2) = KEavg(N2)
(3/2)kT(H2) = (3/2)kT(N2)
Canceling out the common terms:
T(H2) = T(N2)
Therefore, the temperature at which hydrogen molecules (H2) have the same average kinetic energy as nitrogen molecules (N2) is also approximately 635 K (361 °C).