Question

The escape speed from the Moon is much smaller than from Earth, around 2.38 km/s.

At what temperature, in kelvins, would hydrogen molecules (with molar mass of 2.016 g/mol) have an rms speed equal to the Moon’s escape speed?

Answer 1

The temperature at which hydrogen molecules would have an rms speed equal to the Moon's escape velocity can be calculated using the formula for root-mean-square speed. Substituting the values and solving the equation yields a temperature of 501.96 K.

Step-by-step explanation:

The root-mean-square (rms) speed of gas molecules is related to temperature through the equation:

Vrms = √((3kT) / m)

Where:

• Vrms is the root-mean-square speed
• k is the Boltzmann constant (1.38 × 10-23 J/K)
• T is the temperature in kelvin
• m is the molar mass of the gas molecule in kg/mol (converted from g/mol)

To find the temperature at which the rms speed of hydrogen molecules (molar mass = 2.016 g/mol) is equal to the Moon's escape velocity (2.38 km/s), we can rearrange the equation:

T = (Vrms2 * m) / (3k)

Let's substitute the values:

T = (2.38 km/s * (1000 m/km) / s)2 * (2.016 g/mol) / (3 * (1.38 × 10-23 J/K))

Simplifying the equation and calculating, the temperature is approximately 501.96 K.