Question

A hand book states that the solubility of \(R \mathrm{NH}_{2}(g)\) in water at 1 atm and \(25^{\circ} \mathrm{C}\) is \(22.41\) volumes of \(R \mathrm{NH}_{2}(g)\) per volume of water. \(\left(\mathrm{pK}_{b}\right.\) of \(R \mathrm{NH}_{2}=4\) ) Find the max. pOH that can be attained by dissolving \(\mathrm{RNH}_{2}\) in water : (a) 1 (b) 2 (c) 4 (d) 6 .

Short answer

The maximum pOH that can be attained by dissolving RNH2 in water is (c) 4.

Step-by-step solution

Understand the Concept of Solubility in Terms of Volumes

The solubility of RNH2 given as 22.41 volumes means that one volume of water at 1 atm and 25°C can dissolve 22.41 volumes of RNH2 gas.

Calculate the Concentration of RNH2 in Molarity

Assuming 1 volume of water is equivalent to 1 L and the volume of RNH2 gas that dissolves is 22.41 L, we use the molarity equation, concentration (M) = moles of solute/volume of solution (L), and ideal gas law PV = nRT to find the number of moles of RNH2. Calculate the number of moles (n) in 22.41 L at STP, P = 1 atm, V = 22.41 L, R = 0.0821 L atm/mol K, and T = 298 K. The molarity of RNH2 in the solution is n/V where V = 1 L of water.

Apply the Base Dissociation Constant (Kb)

pKb is given as 4, so Kb can be found using the relation pKb = -log(Kb). Calculate Kb from this.

Apply the Dissociation Equilibrium and Calculate pOH

For the base RNH2, the dissociation is RNH2 + H2O ⇌ RNH3+ + OH-. At equilibrium, Kb = [RNH3+][OH-]/[RNH2]. Assuming that the concentration of RNH2 remains nearly the same since it is a weak base, and [RNH3+] and [OH-] are equal, solve for [OH-]. Then calculate pOH using the relation pOH = -log[OH-]. This will give the maximum pOH attainable.

Determine the Maximum pOH

Evaluate the pOH by substituting the values found from the previous steps into pOH = -log[OH-]. The result will indicate the maximum pOH that can be attained by dissolving RNH2 in water.

Key concepts

Solubility in Chemistry

Solubility is a key concept in chemistry that refers to the ability of a substance, the solute, to dissolve in a solvent to form a homogeneous solution. At a given temperature and pressure, solubility is often expressed in terms of the maximum amount of solute that can dissolve in a specific amount of solvent. In physical chemistry, we frequently measure solubility for gases in liquids in terms of volumes. The given exercise illustrates this by stating the solubility of a gas, RNH₂, in water at specific conditions.

Understanding solubility in terms of volumes involves recognizing that '22.41 volumes' signifies that 22.41 parts of RNH₂ gas will dissolve in one part of water under the stated conditions. This is an indirect measure and must be converted to a more standard unit like molarity for further calculations. To improve comprehension, imagine filling a container with one liter of water; this amount of water can dissolve gas that would occupy 22.41 liters under the same conditions.

Base Dissociation Constant

The base dissociation constant (K_b) is a value that quantifies the strength of a base in solution. A large K_b means the base dissociates well into its ions, indicating a strong base, while a small K_b signifies weak dissociation and thus a weak base. The dissociation constant is directly related to the concept of equilibrium, as it represents the ratio of the concentration of the product ions to the undissociated base at equilibrium.

When we have the pK_b value, which is the negative logarithm of K_b, we can calculate K_b with the formula K_b = 10^-pK_b. For example, a pK_b of 4 means K_b = 10^-4. This value is essential in predicting the degree to which a base will dissociate in solution, which affects the pH and pOH level of the solution.

pOH Calculation

pOH is a measure of hydroxide ion (OH^-) concentration in a solution and is an important aspect of understanding the pH scale. It is the negative logarithm of the molar concentration of hydroxide ions. The formula is pOH = -log[OH^-]. By calculating the pOH, we can assess the basicity of a solution; the lower the pOH, the more basic the solution is.

In the context of the exercise, after the base RNH₂ dissociates in water, it will produce OH^- ions. To find the pOH, the concentration of these OH^- ions needs to be calculated first. Then, by applying the formula for pOH, one can find the maximum value of pOH that the solution can attain. Since pOH is often less intuitive than pH, remembering that pH + pOH = 14 can help link pOH to the more commonly discussed pH scale when dealing with aqueous solutions.

Molarity and Concentration

Molarity, a central concept in chemistry, refers to the concentration of a solute in a solution. It is defined as moles of solute per liter of solution (mol/L), and it's a standard unit of concentration in the scientific community. Understanding molarity is crucial for quantitative descriptions of chemical reactions in solution and is essential in the fields of stoichiometry, chemical kinetics, and equilibrium.

To calculate molarity from the given volume of gas solubility, we first need to convert the given volume of the solute gas into moles using the Ideal Gas Law before dividing by the volume of the solution. This step is a common point of confusion and warrants careful attention to unit conversion. Ensuring that the volume of gas is measured at the same conditions as the volume of water is essential for the accurate computation of molarity.