calculate and compare the average gas phase translational energies and rms speeds of HE gas and H2 gas at 20 degrees celsius

To calculate and compare the average gas phase translational energies and root-mean-square (rms) speeds of helium gas (He) and hydrogen gas (H2) at 20 degrees Celsius, we need to use the following formulas:

1. Average translational energy (Eavg) = (3/2) * k * T
2. Root-mean-square speed (vrms) = sqrt((3 * k * T) / m)

Where:
- k is the Boltzmann constant (1.38 x 10^-23 J/K)
- T is the temperature in Kelvin (20 degrees Celsius = 293.15 Kelvin)
- m is the molar mass of the gas (He: 4.003 g/mol, H2: 2.016 g/mol)

Let's calculate and compare the values step by step:

For Helium gas (He):
1. Average translational energy:
Eavg_He = (3/2) * k * T
Eavg_He = (3/2) * (1.38 x 10^-23 J/K) * 293.15 K
Eavg_He ≈ 6.21 x 10^-21 J

2. Root-mean-square speed:
vrms_He = sqrt((3 * k * T) / m)
vrms_He = sqrt((3 * (1.38 x 10^-23 J/K) * 293.15 K) / 4.003 g/mol)
vrms_He ≈ 1,313 m/s

For Hydrogen gas (H2):
1. Average translational energy:
Eavg_H2 = (3/2) * k * T
Eavg_H2 = (3/2) * (1.38 x 10^-23 J/K) * 293.15 K
Eavg_H2 ≈ 6.21 x 10^-21 J

2. Root-mean-square speed:
vrms_H2 = sqrt((3 * k * T) / m)
vrms_H2 = sqrt((3 * (1.38 x 10^-23 J/K) * 293.15 K) / 2.016 g/mol)
vrms_H2 ≈ 1,828 m/s

Comparing the values:

- Average translational energy:
Eavg_He = Eavg_H2 ≈ 6.21 x 10^-21 J

- Root-mean-square speed:
vrms_He ≈ 1,313 m/s
vrms_H2 ≈ 1,828 m/s

Therefore, at 20 degrees Celsius, both helium gas (He) and hydrogen gas (H2) have approximately the same average translational energy but different rms speeds. Hydrogen gas (H2) has a higher rms speed compared to helium gas (He).