Question

Determine the mole fraction of carbon dioxide \(\left(\mathrm{CO}_{2}\right)\) dissolved in water at the surface of water at \(300 \mathrm{~K}\). The mole fraction of \(\mathrm{CO}_{2}\) in air is \(0.005\), and the local atmosphere pressure is \(100 \mathrm{kPa}\).

Short answer

Answer: The mole fraction of CO₂ dissolved in water at 300 K is approximately \(1.63 \times 10^{-4} \text{mol/L}\).

Step-by-step solution

Set up the Henry's Law equation for the given problem

Henry's Law can be written as \(C = k_H * P\), where: -C is the concentration of the dissolved gas in the liquid (in this case, the mole fraction of CO₂ in water) -k_H is the Henry's Law constant (depends on the gas and liquid involved, as well as the temperature) -P is the partial pressure of the gas above the liquid (in this case, the partial pressure of CO₂ above the water) Our goal is to find the mole fraction of CO₂ in water (C), given the mole fraction of CO₂ in air and the local atmospheric pressure.

Calculate the partial pressure of CO₂ above the water

Given the mole fraction of CO₂ in air (0.005) and the local atmospheric pressure (100 kPa), we can determine the partial pressure of CO₂ above the water: Partial pressure of CO₂ = Mole fraction of CO₂ in air × Local atmospheric pressure = \(0.005 \times 100 \text{kPa} = 0.5 \text{kPa}\)

Find the Henry's Law constant for CO₂ in water

Look up the value of the Henry's Law constant (k_H) for CO₂ in water at the given temperature (300 K). In this case, using literature sources, the value of k_H for CO₂ in water at 300 K can be found to be approximately \(3.3 \times 10^{-2} \text{mol/(L∙atm)}\).

Convert the partial pressure from kPa to atm

Before applying Henry's Law equation, we need to convert the partial pressure of CO₂ from kPa to atm, as the Henry's Law constant is given in mol/(L∙atm). The conversion factor is 1 atm = 101.325 kPa: Partial pressure of CO₂ (in atm) = Partial pressure of CO₂ (in kPa) / 101.325 = \(0.5 \text{kPa} / 101.325 \approx 4.93 \times 10^{-3} \text{atm}\)

Determine the mole fraction of CO₂ dissolved in water

Now, we can use the Henry's Law equation to find the mole fraction of CO₂ in water: \(C = k_H * P\) \(C = (3.3 \times 10^{-2} \text{mol/(L∙atm)}) * (4.93 \times 10^{-3} \text{atm})\) \(C \approx 1.63 \times 10^{-4} \text{mol/L}\) The mole fraction of CO₂ dissolved in water at the surface of water at 300 K is approximately \(1.63 \times 10^{-4} \text{mol/L}\).

Key concepts

Mole Fraction

In the study of solutions and mixture compositions, the term mole fraction is a valuable concept. It represents the proportion of a particular component within a mixture, relative to the total amount of all substances present. When the moles of a constituent are divided by the total moles of all components, the result is the mole fraction.

To simplify, consider a mixture of gases, where each gas's mole fraction is the ratio of its moles to the total moles of all gases in the mix. This concept becomes crucial in many calculations, especially when working with laws that describe gas behavior and dissolution, such as Henry’s Law.

For a clear example: if a balloon contains 2 moles of oxygen and 3 moles of nitrogen, the mole fraction of oxygen is calculated as \( \frac{2}{2 + 3} = 0.4 \) and the mole fraction of nitrogen as \( \frac{3}{2 + 3} = 0.6 \). Understanding mole fraction aids in deducing other important quantities like partial pressure and dissolved gas concentration in solutions.

Partial Pressure

The concept of partial pressure is integral to understanding gas behavior, especially in mixtures. It refers to the pressure that one component of a mixture of gases would exert if it alone occupied the entire volume of the mixture at the same temperature.

Building on the previous example of mole fraction, the total pressure of a gas mixture can be distributed among its constituents according to their mole fractions. Dalton's Law of Partial Pressures tells us that in a mixture of gases, the total pressure is the sum of the individual partial pressures of each gas component.

To put this into context with Henry's Law, which we’ve seen in the exercise, the mole fraction of \(CO_2\) in air helps to determine the partial pressure of \(CO_2\) above the water. This partial pressure is critical for calculating how much \(CO_2\) will dissolve in the water under a given set of conditions.

Dissolved Gas Concentration

The dissolved gas concentration is a measure of how much gas is present in a liquid, such as water. This is represented by the amount of gas, in moles, that is dissolved in a certain volume of liquid, and is heavily influenced by the gas's solubility and the liquid's properties.

With Henry's Law, we can predict the dissolved gas concentration based on the partial pressure of the gas above the liquid and the gas's specific solubility in that liquid (expressed through the Henry’s Law constant). The direct proportionality described in Henry’s Law comes in handy when trying to understand how factors like pressure and temperature might affect the amount of gas that dissolves in a liquid.

In the context of our solved problem, by determining the partial pressure of \(CO_2\) and using the Henry's Law constant, we were able to successfully calculate the concentration of \(CO_2\) dissolved in water. This calculation is important in environmental studies, beverage carbonation processes, and many other applications in science and industry.