Final answer:
The temperature at which hydrogen molecules with a molar mass of 2.016 g/mol have a root-mean-square velocity of 193 m/s is approximately 300 K.
Step-by-step explanation:
The question is asking for the temperature at which hydrogen molecules (H₂) with a molar mass of 2.016 g/mol have a root-mean-square velocity (Urms) of 193 m/s. To find the temperature, we can use the equation for the root-mean-square speed of a gas, which is derived from the kinetic theory of gases:
Urms = √(3RT/M)
Where Urms is the root-mean-square velocity, R is the universal gas constant (8.314 J/mol·K), T is the temperature in Kelvin, and M is the molar mass in kilograms per mole. We need to convert the molar mass from grams to kilograms, so M = 2.016 x 10⁻³ kg/mol.
Solving for T, we have:
T = (Urms² × M) / (3R)
T = (193² × 2.016 x 10⁻³ kg/mol) / (3 x 8.314 J/mol·K)
After performing the calculation:
T ≈ 300 K