Problem 29 - Chemistry and Chemical Reactivity

Question

Assume you mix \(100.0 \mathrm{mL}\) of \(0.200 \mathrm{M} \mathrm{CsOH}\) with \(50.0 \mathrm{mL}\) of \(0.400 \mathrm{M} \mathrm{HCl}\) in a coffee-cup calorimeter. The following reaction occurs: \(\mathrm{CsOH}(\mathrm{aq})+\mathrm{HCl}(\mathrm{aq}) \rightarrow \mathrm{CsCl}(\mathrm{aq})+\mathrm{H}_{2} \mathrm{O}(\ell)\) The temperature of both solutions before mixing was \(22.50^{\circ} \mathrm{C},\) and it rises to \(24.28^{\circ} \mathrm{C}\) after the acid-base reaction. What is the enthalpy change for the reaction per mole of CsOH? Assume the densities of the solutions are all \(1.00 \mathrm{g} / \mathrm{mL}\) and the specific heat capacities of the solutions are \(4.2 \mathrm{J} / \mathrm{g} \cdot \mathrm{K}\)

Step-by-Step Solution

Answer

The enthalpy change per mole of CsOH is \(-56.13 \mathrm{kJ/mol}\).

Step 1: Calculate Total Volume and Mass

First, let's find the total volume and mass of the mixed solutions. Since the densities of the solutions are given as \(1.00 \mathrm{g} / \mathrm{mL}\), the volume (in mL) is equal to the mass (in grams).1. Volume of \(\mathrm{CsOH}\) solution = \(100.0 \mathrm{mL}\)2. Volume of \(\mathrm{HCl}\) solution = \(50.0 \mathrm{mL}\)Therefore, the total volume is:\[ 100.0 \mathrm{mL} + 50.0 \mathrm{mL} = 150.0 \mathrm{mL} \]Thus, the total mass of the solution is:\[ 150.0 \mathrm{g} \]

Step 2: Calculate Temperature Change

Next, determine the temperature change \(\Delta T\) for the system:Initial temperature of both solutions is \(22.50^{\circ} \mathrm{C}\) and the final temperature is \(24.28^{\circ} \mathrm{C}\). Hence,\[ \Delta T = 24.28^{\circ} \mathrm{C} - 22.50^{\circ} \mathrm{C} = 1.78^{\circ} \mathrm{C} \]

Step 3: Calculate Heat Absorbed

Using the equation for heat absorption: \[ q = mc\Delta T \]where:- \( m = 150.0 \mathrm{g} \) (the total mass),- \( c = 4.2 \mathrm{J} / \mathrm{g} \cdot \mathrm{K} \) (specific heat capacity),- \( \Delta T = 1.78 \mathrm{K} \).Substitute the values:\[ q = 150.0 \mathrm{g} \times 4.2 \mathrm{J} / \mathrm{g} \cdot \mathrm{K} \times 1.78 \mathrm{K} = 1122.6 \mathrm{J} \]

Step 4: Calculate Moles of Reactants

To find the enthalpy change per mole of \(\mathrm{CsOH}\), we first calculate the limiting reactant:Moles of \(\mathrm{CsOH} = 100.0 \mathrm{mL} \times 0.200 \mathrm{mol/L} = 0.0200 \mathrm{mol} \)Moles of \(\mathrm{HCl} = 50.0 \mathrm{mL} \times 0.400 \mathrm{mol/L} = 0.0200 \mathrm{mol} \)Both have equal moles, thus they react completely, and either can be considered as the limiting reactant.

Step 5: Calculate Enthalpy Change per Mole

The enthalpy change \(\Delta H\) per mole of \(\mathrm{CsOH}\) can be found by dividing the total heat absorbed by the number of moles of \(\mathrm{CsOH}\) (since one mole of \(\mathrm{CsOH}\) reacts with one mole of \(\mathrm{HCl}\)):\[ \Delta H = -\frac{q}{\text{moles of } \mathrm{CsOH}} = -\frac{1122.6 \mathrm{J}}{0.0200 \mathrm{mol}} = -56130 \mathrm{J/mol} = -56.13 \mathrm{kJ/mol} \]The negative sign indicates that the reaction is exothermic.

Key Concepts

Heat Absorption

Heat absorption is a key concept in chemistry that deals with how heat is taken in during a process. In reactions like the mixing of solutions in a calorimeter, heat absorption is indicated by a temperature change. When two solutions react, they can either absorb heat from their surroundings or release heat into their surroundings. In this exercise, the temperature increase from the initial to final state demonstrates the heat absorbed during the reaction.

To quantify heat absorption, we use the formula:

• \( q = mc\Delta T \)

where:

• \( q \) is the heat absorbed in Joules,
• \( m \) is the mass of the solution in grams,
• \( c \) is the specific heat capacity in J/g·K,
• \( \Delta T \) is the change in temperature in Kelvin or Celsius.

In the calorimetry experiment you're looking at, the total heat absorbed is 1122.6 J, calculated using these components.

Acid-Base Reaction

An acid-base reaction is a type of chemical reaction that involves the exchange of protons between reactants. In this exercise, the acid-base reaction occurs between cesium hydroxide (CsOH), a base, and hydrochloric acid (HCl), an acid. The chemical equation is:

• \[\mathrm{CsOH}(\mathrm{aq}) + \mathrm{HCl}(\mathrm{aq}) \rightarrow \mathrm{CsCl}(\mathrm{aq}) + \mathrm{H}_2\mathrm{O}(\ell)\]

This reaction is a classic example where an acid reacts with a base to produce salt and water.

• The reactants, CsOH and HCl, both dissolve in water to yield ions.
• CsOH provides OH⁻ ions, and HCl contributes H⁺ ions.
• These ions combine to form water (H₂O), releasing energy in the form of heat.

The heat evolved indicates the reaction is exothermic, as shown by the temperature rise seen in the exercise.

Specific Heat Capacity

Specific heat capacity is a measure of how much heat energy is required to raise the temperature of a substance. It's specific to different materials and is represented by the letter \( c \). In this exercise, water, the primary solvent, has a specific heat capacity of \( 4.2 \text{ J/g·K} \).

This means that it takes 4.2 joules of energy to increase the temperature of 1 gram of the solution by 1 degree Celsius. The specific heat capacity is crucial for understanding how substances interact with heat during chemical reactions.

• Substances with a high specific heat capacity absorb more heat before they can change temperature significantly.
• Conversely, those with low specific heat capacities heat up quickly with little heat.

Thus, the specific heat capacity informs the calorimetry calculations needed to determine the heat absorbed or released in reactions.

Calorimetry Experiment

Calorimetry is an experimental technique used to measure the heat absorbed or released during chemical reactions. Typically, this involves using devices called calorimeters. In the given example, a coffee-cup calorimeter is used.

The basic principle of calorimetry is to measure the change in temperature of a solution due to a reaction.

• A coffee-cup calorimeter is simple and consists of a Styrofoam cup, thermometer, and a lid.
• This setup allows scientists to isolate a reaction to monitor the heat changes without much heat loss to the environment.

In the exercise, the calorimeter recorded a temperature increase from 22.50°C to 24.28°C due to the acid-base reaction. This temperature change helps calculate the heat absorbed (or released) using the heat equation \( q = mc\Delta T \).

Calorimetry is essential beyond basic studies, aiding in understanding thermochemistry and energy changes in chemical reactions.