A "rule of thumb" in planetary science is that a planet can hold on to a gas for the age of the Solar System if the average speed of molecules in the gas is less than one-sixth the escape velocity of the planet (this takes into account the fact that molecules high in an atmosphere are influenced by a smaller pull from gravity ) . The average speed of a molecule depends on the gas temperature and the mass of the molecule according to the following equation: $V_{g a s}=157 \frac{\mathrm{m}}{\mathrm{sec}} \sqrt{\frac{\text { temperature }}{\text { molecule's mass }}}$ The molecular mass used in this formula is determined by adding up the atomic masses of its constituents, so hydrogen molecules $\left(\mathrm{H}_{2}\right)$ have mass $2,$ nitrogen molecules $\left(\mathrm{N}_{2}\right)$ have mass 28
a. What is the average speed of a hydrogen molecule in Earth's atmosphere if the gas temperature is 300 K? The Earth's escape velocity is $11,200 \mathrm{m} / \mathrm{sec}$. Can the Earth hold on to a hydrogen atmosphere?
b. What is the average speed of a nitrogen molecule in Earth's atmosphere? Can the Earth maintain a nitrogen atmosphere?
c. What is the average speed of a hydrogen molecule on Jupiter, where the temperature is $160 \mathrm{K}$ ? Jupiter's escape velocity is $59,500 \mathrm{m} /$ sec. Can Jupiter hold on to a hydrogen atmosphere?
d. If Jupiter were placed at Mercury's distance from the Sun, where the temperature would rise to about $450 \mathrm{K}$, would it be able to hold on to its hydrogen atmosphere?