Question

The electrostriction of ions in water. Formic acid (HCOOH) at infinite dilution in water at \(T=298.15 \mathrm{~K}\) has partial molar volume \(v=34.69 \mathrm{~cm}^{3} \mathrm{~mol}^{-1}\). The change in volume upon ionization is \(\Delta v=-8.44 \mathrm{~cm}^{3} \mathrm{~mol}^{-1}\). If this \(\Delta v\) is distributed uniformly among the first-shell water molecules, how much volume reduction can be attributed to each first-shell water molecule?

Short answer

-2.11 cm\textsuperscript{3} per water molecule.

Step-by-step solution

- Understanding the Given Data

Formic acid (HCOOH) at infinite dilution in water has a partial molar volume of 34.69 cm\textsuperscript{3} mol\textsuperscript{-1}. The change in volume upon ionization, \(\Delta v\), is \(-8.44\) cm\textsuperscript{3} mol\textsuperscript{-1}. We need to determine the volume reduction per first-shell water molecule due to ionization.

- Understanding the First-Shell Water Molecule Concept

The term 'first-shell water molecules' refers to the water molecules that are directly interacting with the ionized formic acid molecules. We must determine how many water molecules typically constitute the first hydration shell around an ion.

- Estimating the Number of First-Shell Water Molecules

Assume a typical first hydration shell contains approximately 4 to 6 water molecules for small ions. For this problem, let's take 4 water molecules as a common estimate for simplicity.

- Calculate the Volume Reduction per Water Molecule

The total volume reduction \(\Delta v \) must be distributed uniformly among the first-shell water molecules. Using the estimate of 4 water molecules, calculate the volume reduction per water molecule.\[ \text{Volume reduction per water molecule} = \frac{\Delta v}{\text{Number of first-shell water molecules}} = \frac{-8.44 \text{ cm}^{3} \text{ mol}^{-1}}{4} = -2.11 \text{ cm}^{3} \text{ mol}^{-1} \]

Key concepts

partial molar volume

Partial molar volume is an important concept in physical chemistry. It represents the change in the volume of a solution when an additional mole of a substance is added, keeping pressure and temperature constant. For formic acid (HCOOH) in water at infinite dilution at 298.15 K, the partial molar volume is given as 34.69 cm³/mol.

The idea is to see how much volume one mole of formic acid occupies when it is infinitely diluted in water. This means that interactions between formic acid molecules are negligible. The partial molar volume can tell us a lot about how substances will behave in a solution.

Partial molar volumes can change with temperature and pressure. However, for many practical calculations, we consider them at constant temperature and pressure, just like in the given problem.

hydration shell

The hydration shell describes the layer of water molecules that surrounds an ion or a polar molecule in a solution. When formic acid ionizes in water, water molecules arrange themselves around the ions to form a hydration shell. This occurs due to interactions between the water molecules and the ion.

The first hydration shell consists of the water molecules that are directly attached to the ion. The number of molecules in this shell can vary, but usually ranges from 4 to 6 for small ions.

This concept is key for explaining electrostriction, where the water molecules come closer together due to the ion's electric field, leading to a volume reduction. Understanding the hydration shell helps us grasp how ions affect the water structure and explain changes in the physical properties of solutions.

volume reduction per molecule

Electrostriction causes a reduction in the volume of the solution because the ions' electric fields pull the water molecules closer. The problem states that ionizing formic acid results in a volume decrease of -8.44 cm³/mol.

To find the volume reduction per molecule, we assume that these changes only affect the water molecules in the first hydration shell. By estimating there are 4 water molecules in this shell, the volume reduction can be distributed among them.

To calculate:

• Volume reduction per molecule = Total volume change ÷ Number of first-shell molecules
• = -8.44 cm³/mol ÷ 4
• = -2.11 cm³/mol per water molecule

This calculation helps us understand the impact of ionization on each water molecule closely interacting with the ion. It also ties back to the importance of knowing partial molar volumes and the structure of the hydration shell.