Question

Most fish need at least \(4 \mathrm{ppm}\) dissolved \(\mathrm{O}_{2}\) in water for survival. (a) What is this concentration in \(\mathrm{mol} / \mathrm{L}\) ? (b) What partial pressure of \(\mathrm{O}_{2}\) above water is needed to obtain \(4 \mathrm{ppm} \mathrm{O}_{2}\) in water at \(10^{\circ} \mathrm{C}\) ? (The Henry's law constant for \(\mathrm{O}_{2}\) at this temperature is \(1.71 \times 10^{-3} \mathrm{~mol} / \mathrm{L}\)-atm.)

Short answer

The concentration of \(\mathrm{O}_{2}\) in water is \(1.25\times 10^{-4}\, \mathrm{mol\,O}_{2}/\mathrm{L}\), and the partial pressure of \(\mathrm{O}_{2}\) above the water needed to obtain a concentration of \(4\, \mathrm{ppm}\) at \(10^{\circ}\, \mathrm{C}\) is approximately \(0.073\, \mathrm{atm}\).

Step-by-step solution

Convert ppm to mol/L

First, we need to convert the given concentration of dissolved oxygen from parts per million (ppm) to moles per liter (mol/L). Given that the molecular mass of O₂ is 32 g/mol, we can first convert ppm to g/L and then g/L to mol/L. Given: 4 ppm O₂ 1 ppm is equal to 1 mg/L, so we have: 4 ppm O₂ = 4 mg/L O₂ Next, we convert to grams: \(4 \, \mathrm{mg}=0.004 \, \mathrm{g} \, \mathrm{O}_{2}\) Finally, we convert from grams to moles, using the molar mass of \(\mathrm{O}_{2}\): \[\frac{0.004\, \mathrm{g\, O }_{2}}{32 \, \mathrm{g/mol}} \approx 1.25\times 10^{-4}\, \mathrm{mol\,O}_{2}\] So, the concentration of \(\mathrm{O}_{2}\) in water is \(1.25\times 10^{-4}\, \mathrm{mol\,O}_{2}/\mathrm{L}\).

Determine the partial pressure of O₂ using Henry's law

Now, we need to find the partial pressure of O₂ above the water needed to obtain the concentration we just calculated. We will use Henry's law: \[c = kH \cdot p\] where \(c\) is concentration in mol/L, \(kH\) is Henry's law constant in mol/L-atm, and \(p\) is the partial pressure of O₂ in atm. Here, we have the concentration \(c = 1.25\times 10^{-4}\, \mathrm{mol\,O}_{2}/\mathrm{L}\) and the Henry's law constant for \(\mathrm{O}_{2}\), \(kH = 1.71 \times 10^{-3}\, \mathrm{mol\,L^{-1}\,atm^{-1}}\). We want to find \(p\). Rearranging Henry's law formula, we obtain: \[p = \frac{c}{kH}\] Plugging in the values we know: \[p = \frac{1.25\times 10^{-4}\, \mathrm{mol\,O}_{2}/\mathrm{L}}{1.71\times 10^{-3} \, \mathrm{mol\,L^{-1}\,atm^{-1}}}\] \[p \approx 0.073\, \mathrm{atm}\] So, the partial pressure of \(\mathrm{O}_{2}\) above the water needed to obtain a concentration of \(4\, \mathrm{ppm}\) at \(10^{\circ}\, \mathrm{C}\) is approximately \(0.073\, \mathrm{atm}\).

Key concepts

Dissolved Oxygen

Dissolved oxygen (DO) is an important measure of the quality of water, often required for the survival of aquatic life. For instance, many fish species need a minimum concentration of 4 ppm (parts per million) of dissolved oxygen. This value indicates the amount of oxygen gas that has been absorbed into the water. Understanding dissolved oxygen is crucial because it supports the respiratory needs of fish and other aquatic organisms.

It’s essential to grasp that dissolved oxygen levels can fluctuate due to several factors like temperature, salinity, and pressure. Cooler temperatures typically allow more oxygen to dissolve, making regions or habitats with cooler water potentially richer in oxygen. Conversely, warmer waters may possess lower levels of dissolved oxygen, stressing aquatic life. Human activities can also impact these levels through pollution and water management practices. Regularly monitoring dissolved oxygen helps ensure healthy ecosystems and can help identify pollution issues early on.

When dealing with concentrations like 4 ppm, scientists often need to convert this to different units for more precise scientific analysis, which leads us to concentration conversions.

Partial Pressure

Partial pressure is a concept in chemistry that helps explain how gases behave in mixtures. Each gas in a mixture exerts its own pressure called the partial pressure. In this context, we're interested in the partial pressure of oxygen (O₂) in the air above the water. This pressure determines how much oxygen will dissolve in the water, which can then be used by aquatic organisms.

Henry's Law is pivotal in understanding the relationship between the partial pressure of a gas and its concentration in a solution. The law states that at a constant temperature, the amount of gas that dissolves in a liquid is directly proportional to the pressure of that gas above the liquid. Hence, knowing the partial pressure of oxygen is essential because it allows us to calculate how much oxygen is dissolved in the water using the formula:

\[ c = k_H \cdot p \]

Where:

• \( c \) is the concentration of the dissolved gas (mol/L)
• \( k_H \) is the Henry’s Law constant specific to the gas in question and dependent on temperature
• \( p \) is the partial pressure of the gas (atm)

By rearranging this formula, one can find the partial pressure needed to achieve a desired gas concentration in water.

Concentration Conversion

Converting concentrations from one unit to another is an essential skill in chemistry. In our example, transforming 4 ppm of dissolved oxygen to molarity (mol/L) is necessary for scientific calculations. But what does ppm mean? It stands for parts per million, which is a ratio of one part of substance to a million parts of the whole.

Since 1 ppm in water is equivalent to 1 mg/L, we start by converting this to grams per liter (g/L). For oxygen, 4 ppm translates to 4 mg/L, which is the same as 0.004 g/L. Next, using the molecular mass of oxygen (32 g/mol), we convert this value to moles per liter (mol/L), the preferred unit for laboratory calculations:

\[\frac{0.004}{32} \approx 1.25 \times 10^{-4} \text{ mol/L}\]

So, the molarity of dissolved oxygen at 4 ppm is approximately \(1.25 \times 10^{-4} \text{ mol/L}\). Being comfortable with these conversions boosts a student's ability to analyze data and apply theories to practical situations.