Question

if each of the gases below are at the same temperature, which has the greatest average kinetic energy?
oxygen (o₂)
argon (ar)
methane (ch₄)
hydrogen (h₂)
all have the same average kinetic energy

if each of the gases below are at the same temperature, which has the greatest root mean square velocity?
oxygen (o₂)
argon (ar)
all have the same root mean square velocity
hydrogen (h₂)
methane (ch₄)

Explanation:

• For the first question: According to the kinetic - molecular theory of gases, the average kinetic energy of gas molecules is given by the formula \(E_{k}=\frac{3}{2}kT\), where \(k\) is the Boltzmann constant and \(T\) is the temperature. Since all the gases are at the same temperature \(T\), the average kinetic energy depends only on \(T\).
• For the second question: The root - mean - square velocity of gas molecules is given by the formula \(v_{rms}=\sqrt{\frac{3RT}{M}}\), where \(R\) is the universal gas constant, \(T\) is the temperature, and \(M\) is the molar mass of the gas. At the same temperature \(T\), the gas with the smallest molar mass \(M\) will have the largest \(v_{rms}\). The molar masses are: \(M(O_{2}) = 32\space g/mol\), \(M(Ar)=40\space g/mol\), \(M(CH_{4}) = 16\space g/mol\), \(M(H_{2})=2\space g/mol\).

Answer:

• First question: all have the same average kinetic energy
• Second question: hydrogen (\(H_{2}\))