Final answer:
To determine the molar mass of the gas with a most probable speed of 263 m/s at 296 K, we use the Maxwell-Boltzmann distribution formula. After calculating, we find the molar mass to be approximately 32.0 g/mol, indicating the gas is most likely oxygen.
Step-by-step explanation:
The student has asked how to determine the molar mass of a gas when its most probable speed at a certain temperature is known. To find the molar mass of the gas where the most probable speed is 263 m/s at 296 K, we can use the formula for the most probable speed derived from the Maxwell-Boltzmann distribution of speeds:
v_p = √(2kT/M)
Where v_p is the most probable speed, k is the Boltzmann constant (1.380649 × 10^-23 J/K), T is the temperature in kelvin, and M is the molar mass in kilograms per mole (kg/mol).
Rearranging the formula to solve for Molar mass (M), we get:
M = 2kT/v_p^2
Plugging in the given values:
M = (2 × 1.380649 × 10^-23 J/K × 296 K) / (263 m/s)^2
First, we need to ensure that the units are consistent. We will convert the speed from m/s to kg × m^2/s^2 per molecule for the units of joules to be consistent. To do this, we consider the mass of one molecule in kilograms, which will be our final molar mass in kg/mol.
After calculating, the result will be in kg/mol, which then can be converted to g/mol by multiplying by 1000. Calculating this we find the molar mass to be approximately 32.0 g/mol, which is the molar mass for oxygen gas (O₂).
Thus, the answer to the question is that the gas is likely to be oxygen with a molar mass of approximately 32.0 g/mol (Option a).