Problem 27 - Chemistry: Introducing Inorganic, Organic and Physical Chemistry

Question

A chemist is trying to measure the enthalpy change for the hydration reaction: \\[ \mathrm{Na}_{2} \mathrm{CO}_{3}(\mathrm{s}) \rightarrow \mathrm{Na}_{2} \mathrm{CO}_{3} \cdot 10 \mathrm{H}_{2} \mathrm{O}(\mathrm{s}) \\] Calibration of a solution calorimeter shows that 357.9 J of heat raises the temperature by \(1.00 \mathrm{K}\). When \(2.500 \mathrm{g}\) of anhydrous sodium carbonate, \(\mathrm{Na}_{2} \mathrm{CO}_{3}\), was dissolved in \(100.00 \mathrm{cm}^{3}\) of water, the temperature increased by \(1.550 \mathrm{K}\). On dissolving \(3.500 \mathrm{g}\) of sodium carbonate decahydrate, \(\mathrm{Na}_{2} \mathrm{CO}_{3} \cdot 10 \mathrm{H}_{2} \mathrm{O}\) in \(100.00 \mathrm{cm}^{3}\) of water, the temperature decreased by \(2.310 \mathrm{K}\) Calculate enthalpy change of hydration of sodium carbonate. (Section 13.5 )

Step-by-Step Solution

Answer

The enthalpy change for hydration of sodium carbonate is approximately -58.55 kJ/mol.

Step 1: Calculate Heat Absorbed by Water

First, calculate the heat absorbed by the water when 2.500 g of anhydrous Na₂CO₃ is dissolved. Use the formula for heat transfer \[ q = C \times \Delta T \] where \( C \) is the heat capacity of the solution calorimeter (357.9 J/K) and \( \Delta T \) is the temperature change (1.550 K). \[ q = 357.9 \, \text{J/K} \times 1.550 \, \text{K} = 554.745 \, \text{J} \]

Step 2: Calculate Heat Absorbed by Decahydrate

Next, calculate the heat absorbed by the water when 3.500 g of Na₂CO₃ · 10H₂O is dissolved, causing a temperature decrease of 2.310 K.\[ q = 357.9 \, \text{J/K} \times (-2.310 \, \text{K}) = -826.749 \, \text{J} \]

Step 3: Calculate Enthalpy Change of Hydration

The enthalpy change of the hydration reaction can be found by taking the difference between the two heats calculated. The reaction goes from anhydrous to hydrated form, releasing heat in the decahydrate dissolution and absorbing heat in the anhydrous dissolution.\[ \Delta H = q_{\text{decahydrate}} - q_{\text{anhydrous}} = -826.749 \, \text{J} - 554.745 \, \text{J} = -1381.494 \, \text{J} \] To convert this to molar enthalpy change (J/mol), you need to divide by the moles of 2.500 g of anhydrous Na₂CO₃. Use the molar mass of Na₂CO₃ (105.99 g/mol).\[ \text{Moles of Na}_2\text{CO}_3 = \frac{2.500 \, \text{g}}{105.99 \, \text{g/mol}} \approx 0.0236 \, \text{mol} \]\[ \Delta H_{\text{molar}} = \frac{-1381.494 \, \text{J}}{0.0236 \, \text{mol}} \approx -58545 \, \text{J/mol} \approx -58.55 \, \text{kJ/mol} \]

Key Concepts

Hydration Reaction

In chemistry, a hydration reaction involves the addition of water molecules to a substance without the substance decomposing. In the case of sodium carbonate decahydrate, the hydration reaction is essential because it involves converting an anhydrous (without water) form of sodium carbonate to a hydrated form, which contains water bound within its structure.

This transformation can be represented by the chemical equation: \[ \text{Na}_2\text{CO}_3(\text{s}) \rightarrow \text{Na}_2\text{CO}_3 \cdot 10 \, \text{H}_2\text{O}(\text{s}) \] When this reaction occurs, energy, in the form of heat, is either absorbed or released.

• **Endothermic Reaction**: If the reaction absorbs heat, it results in a temperature drop in the surrounding environment.
• **Exothermic Reaction**: Conversely, if the reaction releases heat, there's a rise in temperature in the surroundings.

In the provided exercise, the hydration of sodium carbonate is essential, as it helps us understand whether the reaction is primarily endothermic or exothermic.

Calorimetry

Calorimetry is a technique used to measure the amount of heat exchanged in a chemical reaction. The practice involves using a device known as a calorimeter, which insulates the reaction to measure any temperature changes accurately. In this exercise, we used a solution calorimeter, specifically calibrated to determine the enthalpy change of a reaction.

The calorimeter lets us measure the heat absorbed or released by observing the temperature change in its content. When using calorimetry:

• The **heat capacity** of the calorimeter must be known so that we can accurately calculate the heat exchanged.
• The formula used is: \[ q = C \times \Delta T \] where \( q \) is the heat absorbed or released, \( C \) stands for the heat capacity, and \( \Delta T \) represents the temperature change.

This crucial setup allows chemists to obtain precise measurements and ensures that experimental conditions align well with theoretical predictions.

Heat Capacity

Heat capacity is a property that defines how much heat is required to change the temperature of a substance by a certain amount. For a calorimeter, knowing its heat capacity is essential for accurate measurements, as it impacts how the instrument detects and calculates heat changes.

In this scenario, we have a calorimeter with a heat capacity of 357.9 J/K, meaning that adding 357.9 joules of heat will increase its temperature by 1 K.

• When it comes to calorimetry:
  • A higher heat capacity means the substance can absorb more heat without a significant increase in temperature.
  • A lower heat capacity results in a larger temperature change for the same amount of heat.

Since heat capacity is an intrinsic property, it dictates every calculation in calorimetry, affecting the interpretation of any thermodynamic data collected during the experiments.

Sodium Carbonate Decahydrate

Sodium carbonate decahydrate (\( \text{Na}_2\text{CO}_3 \cdot 10 \, \text{H}_2\text{O} \)) is the hydrated form of sodium carbonate. It contains ten water molecules coordinated within its crystal structure. This specific form of sodium carbonate is often involved in experiments where understanding the role of water of crystallization is vital.

The use of sodium carbonate decahydrate in experiments often highlights the difference between an anhydrous and a hydrated substance. In the provided problem:

• The decahydrate dissolving in water resulted in a decrease in temperature, demonstrating the enthalpy change during the hydration process.
• The process shed light on how heat exchange is involved when a salt transforms from its anhydrous form to a hydrated form.

Understanding sodium carbonate decahydrate allows for a deeper insight into hydration reactions and enthalpy changes, particularly in the framework of solubility and endothermic/exothermic reactions.