Question

The Henry's law constant for the solubility of helium gas in water is\(3.8 \times 10^{-4} \mathrm{M} / \mathrm{atm}\) at \(25^{\circ} \mathrm{C}\). (a) Express the constant for the solubility of helium gas in \(M / \mathrm{mm} \mathrm{Hg}\). (b) If the partial pressure of \(\mathrm{He}\) at \(25^{\circ} \mathrm{C}\) is \(293 \mathrm{~mm} \mathrm{Hg}\), what is the concentration of dissolved He in \(\mathrm{mol} / \mathrm{L}\) at \(25^{\circ} \mathrm{C} ?\) (c) What volume of helium gas can be dissolved in \(10.00 \mathrm{~L}\) of water at \(293 \mathrm{~mm} \mathrm{Hg}\) and \(25^{\circ} \mathrm{C}\) ? (Ignore the partial pressure of water.)

Short answer

Question: Convert the given Henry's law constant for helium (3.8 x 10^-4 M/atm) to M/mmHg, calculate the concentration of dissolved helium in mol/L at 25°C and 293 mmHg, and determine the volume of helium gas that can dissolve in 10 L of water at these conditions. Answer: The converted Henry's law constant for helium is 5.0 x 10^-7 M/mmHg. The concentration of dissolved helium at 25°C and 293 mmHg is 1.47 x 10^-4 M. Approximately 1.63 x 10^-2 L of helium gas can dissolve in 10 L of water at the given conditions.

Step-by-step solution

Part (a): Convert Henry's Law Constant

We need to convert the given Henry's law constant for helium (in M/atm) to M/mmHg. We can do this using the conversion factor: 1 atm = 760 mmHg. So, \(K_{\text{M/atm}} = 3.8 \times 10^{-4}\) M/atm. To convert this value to M/mmHg, multiply it by the conversion factor: \(K_{\text{M/mmHg}} = 3.8 \times 10^{-4} \frac{\text{M}}{\text{atm}} \times \frac{1 \text{ atm}}{760 \text{ mmHg}}\) Calculate the new value: \(K_{\text{M/mmHg}} = 5.0 \times 10^{-7} \text{ M/mmHg}\)

Part (b): Calculate the concentration of dissolved helium

Now, we will use the Henry's law constant in M/mmHg and the given partial pressure of helium to calculate the concentration of dissolved helium in mol/L: \(C = K_{\text{M/mmHg}} \times P_\text{He}\) Substitute the given values: \(C = 5.0 \times 10^{-7} \text{ M/mmHg} \times 293 \text{ mmHg}\) Calculate the concentration: \(C = 1.47 \times 10^{-4} \text{ M}\) The concentration of dissolved helium in the water is \(1.47 \times 10^{-4} \text{ M}\) at \(25^{\circ} \mathrm{C}\).

Part (c): Calculate the volume of dissolved helium gas

To find the volume of helium gas that can dissolve in \(10.00 \text{ L}\) of water, we must first calculate the amount of helium (in moles) that can dissolve in that volume of water. To do this, multiply the concentration by the volume of water: \(n_\text{He} = C \times V\) Substitute the concentration and volume: \(n_\text{He} = 1.47 \times 10^{-4} \text{ M} \times 10.00 \text{ L}\) Calculate the number of moles: \(n_\text{He} = 1.47 \times 10^{-3} \text{ mol}\) Next, we will use the ideal gas law to find the volume of helium gas at the given conditions (\(25^{\circ} \mathrm{C}\), \(293 \text{ mmHg}\)). \(PV = nRT\) Rearrange to solve for the volume (note that we have to convert the temperature to Kelvin): \(V = \frac{nRT}{P} = \frac{1.47 \times 10^{-3} \text{ mol} \times 0.0821 \frac{\text{L} \cdot \text{atm}}{\text{mol} \cdot \text{K}} \times (25 + 273.15) \text{ K}}{293 \text{ mmHg} \times \frac{1 \text{ atm}}{760 \text{ mmHg}}}\) Calculate the volume of helium: \(V \approx 1.63 \times 10^{-2} \text{ L}\) So, approximately \(1.63 \times 10^{-2} \text{ L}\) of helium gas can dissolve in \(10.00 \text{ L}\) of water at \(293 \mathrm{~mm} \mathrm{Hg}\) and \(25^{\circ} \mathrm{C}\).

Key concepts

Gas Solubility in Liquids

Understanding the principles of gas solubility in liquids is crucial for grasping a wide range of scientific and industrial processes, including how gases are dissolved in our blood and beverages like soda. Henry's Law provides a simple way to quantify gas solubility in liquids, stating that at a constant temperature, the amount of dissolved gas is directly proportional to the partial pressure of that gas above the liquid. This relationship is captured by the formula:

\(\text{C} = kP\)
Where \(C\) is the molar concentration of the dissolved gas, \(k\) is the Henry's Law constant specific to the gas and solvent at a given temperature, and \(P\) is the partial pressure of the gas.

In real-life situations, such as scuba diving or the production of carbonated drinks, knowing the solubility of gases can prevent dangerous outcomes like decompression sickness, or provide the right fizziness to a soft drink. When converting units, as we did in step 1 of the solution, it's essential to employ known conversion factors, such as 1 atm being equivalent to 760 mmHg. The converted Henry's Law constant allows us to predict the behavior of gases across a range of pressures commonly found in everyday life.

Chemical Concentration Calculations

The concentration of a chemical in a solution is a fundamental concept in chemistry, detailing the amount of a substance per unit volume. Molarity \(M\), expressed in moles per liter (mol/L), is a commonly used unit to describe chemical concentrations. It is critical for predicting how substances will react together and for calculating the necessary reagents in a reaction.

For instance, step 2 of the provided solution demonstrates the practical use of Henry's Law to calculate the concentration of dissolved helium in water, using the formula:
\(C = K_{\text{M/mmHg}} \times P_{\text{He}}\)
Substituting the value of the Henry's Law constant and the partial pressure of helium gives us the molar concentration of helium. Accurate concentration calculations are essential not only in lab settings for making solutions but also in environmental monitoring, medicine dosage, and industrial processes. Simple multiplication or division can sometimes help convert concentrations between different volume units, which is a minor but crucial aspect of chemical concentration calculations.

Ideal Gas Law

The Ideal Gas Law is a vital equation in chemistry and physics, encapsulating the relationships between pressure (\(P\)), volume (\(V\)), temperature (\(T\)), and moles of gas (\(n\)), under ideal conditions. The law is represented as:
\(PV = nRT\)
Where \(R\) is the universal gas constant with a value of 0.0821 L·atm/mol·K. This equation provides a close approximation of the behavior of real gases at a wide range of temperatures and pressures.

In the solution's step 3, we saw an application of the Ideal Gas Law to find the volume of helium gas that can be dissolved in water. By rearranging the equation, we're able to determine the volume that a known amount of gas will occupy under specific conditions. One should remember to convert all units to the standard system, like converting Celsius to Kelvin for temperature, as we did in the calculation. This equation is crucial for applications across science and engineering, like designing pressurized systems, studying atmospheric conditions, or even calculating the required fuel in space missions.