Problem 24 - Chemistry and Chemical Reactivity

Question

Butane, \(\mathrm{C}_{4} \mathrm{H}_{10}\), has been suggested as the refrigerant in household compressors such as those found in air conditioners. (a) To what extent is butane soluble in water? Calculate the butane concentration in water if the pressure of the gas is 0.21 atm. \(\left(k_{\mathrm{H}}=\right.\) \(\left.0.0011 \mathrm{mol} / \mathrm{kg} \cdot \text { bar at } 25^{\circ} \mathrm{C}\right)\) (b) If the pressure of butane is increased to 1.0 atm, does the butane concentration increase or decrease?

Step-by-Step Solution

Answer

(a) Butane concentration is 0.00023406 mol/kg at 0.21 atm. (b) Concentration increases with 1.0 atm pressure.

Step 1: Understanding Henry's Law

Henry's Law states that the concentration of a gas in a liquid is directly proportional to the partial pressure of the gas above the liquid. Mathematically, this can be expressed as \( C = k_{H} \times P \), where \( C \) is the concentration, \( k_H \) is the Henry's law constant, and \( P \) is the partial pressure of the gas.

Step 2: Convert Pressure Units

Convert the pressure from atm to bar because the Henry's law constant, \( k_H \), is given in terms of bar. 1 atm = 1.01325 bar. Therefore, 0.21 atm = 0.21 \times 1.01325 = 0.2127825 bar.

Step 3: Calculate Butane Concentration (Part a)

Using the converted pressure and Henry's law constant, calculate the concentration of butane in water: \[ C = 0.0011 \text{ mol/kg} \cdot \text{bar} \times 0.2127825 \text{ bar} = 0.00023406 \text{ mol/kg}. \] So, the concentration of butane in water at 0.21 atm pressure is 0.00023406 mol/kg.

Step 4: Determine Concentration Change (Part b)

If the pressure is increased to 1.0 atm, first convert it to bar. 1.0 atm = 1.01325 bar.Recalculate using Henry's Law:\[ C = 0.0011 \text{ mol/kg} \cdot \text{bar} \times 1.01325 \text{ bar} = 0.001114575 \text{ mol/kg}. \]Increasing the pressure from 0.21 atm to 1.0 atm increases the concentration of butane in water from 0.00023406 mol/kg to 0.001114575 mol/kg.

Key Concepts

Gas Solubility

The solubility of a gas in a liquid, such as butane in water, refers to how much of the gas can dissolve in the liquid at a given temperature and pressure. Gas solubility varies widely depending on factors like temperature, pressure, and the nature of the gas and liquid. For example, certain gases are more soluble in water than others due to their particular chemical properties.

In the context of our exercise, butane's solubility is quantified using Henry's Law. According to this principle, the solubility of a gas in a liquid is directly related to the partial pressure of that gas above the liquid. This means that by knowing the partial pressure, one can determine how much butane will dissolve in water at a certain temperature. The proportional relationship also indicates if the partial pressure is low, the solubility will likewise be low, and vice versa.

Commonly, for gases like butane, their low solubility makes them suitable for applications where quick vaporization is desirable, such as in refrigeration. However, their low solubility underscores the need for careful pressure management to ensure effectiveness and safety.

Partial Pressure

The concept of partial pressure is essential when determining how much of a gas will dissolve in a liquid like water. Partial pressure measures the pressure exerted by a single type of gas in a mixture of gases.

In our case, butane's behavior under varying pressures has been examined. At 0.21 atm, the partial pressure allows for the calculation of butane's concentration in the water using the formula prescribed by Henry's Law. This shows butane's concentration in water to be 0.00023406 mol/kg under that pressure.

• Partial pressure directly influences gas solubility: higher partial pressure leads to greater solubility.
• The conversion from atmospheres (atm) to bars is crucial because the Henry's Law constant used here requires pressure in bars.
• When the pressure is raised to 1.0 atm (1.01325 bars), the solubility of butane in water increases, reflecting the increased partial pressure.

Understanding these concepts helps in predicting how changes in pressure impact gas solubility, allowing for proper control over processes involving gas dissolving in liquids.

Chemical Concentration

Chemical concentration refers to the amount of a substance, in this case, butane, present in a given quantity of solvent, which is water here. This is often expressed in terms of molarity, moles per kilogram (mol/kg), or other similar units that denote the amount of solute per unit of solvent.

Using Henry's Law, the calculation of butane's concentration in water provides insight into how much of this gas can dissolve under various pressures. For instance:

• At a pressure of 0.21 atm (converted to 0.2127825 bar), the calculated concentration of butane is 0.00023406 mol/kg.
• If the partial pressure increases to 1.0 atm (1.01325 bar), then the concentration rises to 0.001114575 mol/kg.

This change in concentration illustrates the direct proportionality highlighted by Henry's Law: as pressure increases, so does solubility, assuming constant temperature and solubility constants (k_H).

Grasping these calculations and relationships helps with applying these principles to practical scenarios, such as choosing suitable refrigerants for certain pressure conditions.