Question

Oxygen gas is forced into an aquarium at \(1 \mathrm{~atm}\) and $25^{\circ} \mathrm{C}$, and the oxygen bubbles are observed to rise to the free surface in \(4 \mathrm{~s}\). Determine the penetration depth of oxygen into water from a bubble during this time period.

Short answer

In this problem, we are asked to find the penetration depth of oxygen into water from a bubble as it rises to the surface in 4 seconds. We first convert the given temperature from Celsius to Kelvin, which results in 298.15 K. Then, we estimate the diffusion coefficient of oxygen in water using the Stokes-Einstein equation, obtaining a value of approximately 2.17 x 10^-9 m²/s. Finally, we use Fick's first law of diffusion to calculate the penetration depth, which comes out to be approximately 1.30 x 10^-4 m or 130 µm.

Step-by-step solution

Identify known variables

The given variables are: Pressure (P) of oxygen gas = 1 atm, Temperature (T) = \(25^{\circ} \mathrm{C}\), and time (t) = 4 s.

Convert temperature from Celsius to Kelvin

In order to calculate the diffusion coefficient, we need to convert the given temperature to Kelvin. To do this, simply add 273.15 to the given Celsius temperature: \(T (K) = T (^{\circ}\mathrm{C}) + 273.15 = 25 + 273.15 = 298.15\,\mathrm{K}\)

Estimate the diffusion coefficient of oxygen in water

We can use the Stokes-Einstein equation to estimate the diffusion coefficient (D) of oxygen in water as follows: \(D = \frac{k\,T}{6\,\pi\,\eta\,r}\), where \(k\) is the Boltzmann constant (\(1.38 \times 10^{-23}\,\mathrm{J\cdot K^{-1}}\)), \(T\) is the temperature in Kelvin, \(\eta\) is the dynamic viscosity of the medium (water), and \(r\) is the radius of the diffusing molecule (oxygen). For water at \(25^{\circ}\mathrm{C}\), the dynamic viscosity is approximately \(8.9\times10^{-4}\,\mathrm{Pa\cdot s}\), and for oxygen, the molecular radius can be approximately \(1.4\times10^{-10}\,\mathrm{m}\). Plugging in these values into the Stokes-Einstein equation, we have: \(D = \frac{(1.38 \times 10^{-23}\,\mathrm{J\cdot K^{-1}})(298.15\,\mathrm{K})}{6\,\pi\,(8.9\times 10^{-4}\,\mathrm{Pa\cdot s})(1.4\times10^{-10}\,\mathrm{m})} \approx 2.17\times10^{-9}\,\mathrm{m^2\cdot s^{-1}}\)

Calculate the penetration depth of oxygen into water

We can now use Fick's first law of diffusion to estimate the penetration depth (x) of oxygen into water over the given time period: \(x = \sqrt{2\,D\,t}\), where \(D\) is the diffusion coefficient obtained in the previous step, and \(t\) is the time. Plugging the values into the equation, we have: \(x = \sqrt{2 \times 2.17\times10^{-9}\,\mathrm{m^2\cdot s^{-1}} \times 4\,\mathrm{s}} \approx 1.30\times 10^{-4}\,\mathrm{m}\)

Present the solution

The penetration depth of oxygen into water from a bubble during the time period of 4 seconds is approximately \(1.30\times 10^{-4}\,\mathrm{m}\) or \(130\,\mathrm{µm}\).