In this problem, we are tasked with calculating the bulk modulus of oxygen gas using its mass, volume, and the speed of sound within the gas. We start with the known values: the mass of oxygen is 32 grams, which converts to 0.032 kilograms, and it occupies a volume of 0.0224 cubic meters. The speed of sound in this container is given as 317 meters per second.

To find the bulk modulus (denoted as β), we utilize the relationship between the speed of sound (v), bulk modulus (β), and density (ρ) of the gas. The relevant equation for longitudinal waves in fluids is:

$$ v = \sqrt{\frac{\beta}{\rho}} $$

From this equation, we can rearrange it to express the bulk modulus as:

$$ \beta = v^2 \cdot \rho $$

Next, we need to calculate the density (ρ) of the gas. Density is defined as mass divided by volume:

$$ \rho = \frac{mass}{volume} $$

Substituting the values we have:

$$ \rho = \frac{0.032 \, \text{kg}}{0.0224 \, \text{m}^3} $$

Calculating this gives us a density of approximately 1.43 kilograms per cubic meter.

Now, we can substitute the values of speed of sound and density back into the rearranged equation for bulk modulus:

$$ \beta = (317 \, \text{m/s})^2 \cdot (1.43 \, \text{kg/m}^3) $$

Calculating this yields a bulk modulus of:

$$ \beta \approx 1.44 \times 10^5 \, \text{Pascals} $$

This value represents the bulk modulus of oxygen gas, which is crucial for understanding its compressibility and behavior under pressure. The final answer aligns with the expected results, confirming the calculations are correct.