Introduction:
The de Broglie wavelength is a concept in quantum mechanics that describes the wave-like behavior of particles. It is given by the equation λ = h / p, where λ is the de Broglie wavelength, h is the Planck's constant, and p is the momentum of the particle.

Calculation of Root-Mean-Square Velocity:
To calculate the root-mean-square velocity of hydrogen molecules at a temperature of 20 degrees Celsius, we can use the following formula:

v(rms) = √(3kT / m)

Where v(rms) is the root-mean-square velocity, k is the Boltzmann constant, T is the temperature in Kelvin, and m is the mass of the hydrogen molecule.

Conversion of Temperature to Kelvin:
To convert the temperature from Celsius to Kelvin, we add 273.15 to the given temperature.

T(K) = 20 + 273.15 = 293.15 K

Mass of Hydrogen Molecule:
The mass of a hydrogen molecule, H2, is given by the sum of the masses of two hydrogen atoms.

m(H2) = 2 * m(H)

Where m(H) is the mass of a hydrogen atom.

The mass of a hydrogen atom is approximately 1.00784 atomic mass units (u).

m(H) ≈ 1.00784 u

Calculation of Root-Mean-Square Velocity:
Now, we can substitute the values into the formula for root-mean-square velocity.

v(rms) = √(3kT / m(H2))
= √(3 * 1.380649 × 10^-23 J/K * 293.15 K / (2 * 1.00784 u))
≈ √(1.27864 × 10^-20 J / 0.00201568 kg)
≈ √6.34 × 10^(-18) m^2/s^2

Calculation of de Broglie Wavelength:
Finally, we can calculate the de Broglie wavelength using the root-mean-square velocity.

λ = h / p

The momentum can be calculated using the equation p = mv.

p = m(v(rms))

Substituting the values, we get:

λ = h / (m(v(rms)))
= (6.62607015 × 10^-34 J·s) / (0.00201568 kg * √6.34 × 10^(-18) m^2/s^2)
≈ 0.0000000000000021 m

Conclusion:
Therefore, the de Broglie wavelength corresponding to the root-mean-square velocity of hydrogen molecules at a temperature of 20 degrees Celsius is approximately 0.0000000000000021 meters.