Final answer:
The root-mean-square speed of methane (CH4) gas molecules at a temperature of 505 K is approximately 1603 m/s.
Step-by-step explanation:
The root-mean-square (rms) speed of methane (CH4) gas molecules at a temperature of 505 K can be calculated using the formula K = mv² = 2 kBT, where K is the kinetic energy, m is the molar mass, and T is the temperature in Kelvin.
To convert the temperature to Kelvin, we add 273 to the given temperature of 505 K, giving us 778 K. Next, we need to find the molar mass of methane, which is 16.04 g/mol. We convert this to kg/mol by dividing by 1000, giving us 0.01604 kg/mol.
Now, we can calculate the rms speed using the formula. Plugging in the values, we have:
K = mv² = 2 kBT 0.01604 kg/mol * (urms)² = 2 * (1.38 × 10^-23 J/K) * (778 K)
Simplifying the equation gives us (urms)² = (2 * (1.38 × 10^-23 J/K) * (778 K)) / (0.01604 kg/mol).
Taking the square root of both sides, we find that urms ≈ 1603 m/s. Therefore, the root-mean-square speed of methane gas molecules at a temperature of 505 K is approximately 1603 m/s.