Problem 38 - Chemistry and Chemical Reactivity

Question

You mix \(125 \mathrm{mL}\) of \(0.250 \mathrm{M}\) CsOH with \(50.0 \mathrm{mL}\) of 0.625 M HF in a coffee-cup calorimeter, and the temperature of both solutions rises from \(21.50^{\circ} \mathrm{C}\) before mixing to \(24.40^{\circ} \mathrm{C}\) after the reaction. $$ \mathrm{CsOH}(\mathrm{aq})+\mathrm{HF}(\mathrm{aq}) \rightarrow \mathrm{CsF}(\mathrm{aq})+\mathrm{H}_{2} \mathrm{O}(\ell) $$ What is the enthalpy of 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 of reaction per mole of CsOH is \(-68.2\, \text{kJ/mol}\).

Step 1: Calculate the Total Volume of the Solution

Add the volumes of CsOH and HF solutions together. \[ V_{total} = 125 \text{ mL} + 50.0 \text{ mL} = 175 \text{ mL} \]

Step 2: Determine the Total Mass of the Solution

Since the density is given as \(1.00 \text{ g/mL}\), the total mass \(m\) can be calculated by multiplying the total volume by the density. \[ m = 175 \text{ mL} \times 1.00 \text{ g/mL} = 175 \text{ g} \]

Step 3: Calculate the Temperature Change

Find the difference between the final and initial temperatures. \[ \Delta T = 24.40^{\circ} \text{C} - 21.50^{\circ} \text{C} = 2.90^{\circ} \text{C} \]

Step 4: Calculate the Heat Absorbed by the Solution (q)

Use the formula for heat: \[ q = m \cdot c \cdot \Delta T \] where \(c\) is the specific heat capacity. Substitute the known values:\[ q = 175 \text{ g} \times 4.2 \text{ J/g} \cdot \text{K} \times 2.90 \text{ K} \] Calculate \(q\). \[ q = 2131.5 \text{ J} \]

Step 5: Calculate the Limiting Reactant and Moles of CsOH

Calculate moles of CsOH and HF:\[ \text{moles of CsOH} = 125 \text{ mL} \times 0.250 \text{ M} \times 10^{-3} = 0.03125 \text{ moles} \]\[ \text{moles of HF} = 50.0 \text{ mL} \times 0.625 \text{ M} \times 10^{-3} = 0.03125 \text{ moles} \]Both reactants are present in the same stoichiometric amount; any of them could limit the reaction.

Step 6: Calculate the Enthalpy Change per Mole of CsOH

The reaction occurs stoichiometrically with 0.03125 moles of CsOH. Enthalpy change \(\Delta H\) per mole is given by:\[ \Delta H = \frac{q}{\text{moles of CsOH}} \]Substitute the known values:\[ \Delta H = \frac{2131.5 \text{ J}}{0.03125 \text{ moles}} = 68208 \text{ J/mol} \]Convert to kJ/mol: \[ \Delta H = -68.2 \text{ kJ/mol} \]

Step 7: Remember the Sign Convention

Since the temperature increases, the reaction is exothermic, and the enthalpy should be negative. Therefore, \(\Delta H = -68.2 \text{ kJ/mol}\).

Key Concepts

Limiting Reactant

In a chemical reaction, the limiting reactant is the substance that is entirely consumed first. This component determines the maximum amount of product that can be formed. When you calculate the moles of each reactant, you can compare them to find out which one will run out first. In the given exercise, both Cesium Hydroxide (CsOH) and Hydrofluoric Acid (HF) can potentially limit the reaction because they are present in equal molar amounts (0.03125 moles each). This means either can fully react and become the limiting reagent.
To decide which reactant limits the reaction in more complex situations, you would compare the mole ratio from the balanced equation with the actual mole amounts you calculated. The reactant with a smaller mole-to-coefficient ratio is the limiting reactant. Knowing the limiting reactant is crucial for calculating the theoretical yield of the reaction and for other stoichiometric calculations.

Coffee-Cup Calorimeter

A coffee-cup calorimeter is a simple and convenient device used to measure the heat of chemical reactions, particularly in solutions. Named for its resemblance to a coffee cup, this calorimeter is usually made up of two nested Styrofoam cups, which help to insulate and minimize heat loss to the environment.
The calorimeter measures the temperature change of a solution when a chemical reaction occurs. The formula used is:

• q = m × c × ΔT

where: - q is the heat absorbed or released, - m is the mass of the solution, - c is the specific heat capacity, - ΔT is the temperature change.
In this particular problem, the coffee-cup calorimeter setup allows us to assume that all the heat released by the reaction goes into warming the solution, making calculations more straightforward. Coffee-cup calorimeters are typically used for reactions that take place in aqueous solutions, such as neutralization reactions, where it’s important to understand how much heat is involved.

Stoichiometry

Stoichiometry is an essential part of chemistry that involves calculating the amounts of reactants and products involved in a chemical reaction. It is grounded in the principle that matter cannot be created or destroyed in a chemical reaction, so the number of atoms of each element must be the same on both sides of the equation.
To use stoichiometry, you need a balanced chemical equation. This tells you the ratio in which reactants react and products form. In our reaction:

• CsOH(aq) + HF(aq) → CsF(aq) + H₂O(l)

the stoichiometric ratio of to CsOH to HF is 1:1. This tells us for every one mole of CsOH, one mole of HF is required.
Using stoichiometric calculations, you can determine the amount of each reactant needed to produce a desired amount of product, or the amount of product that can be made from given amounts of reactants. This concept is vital when performing chemical reactions in laboratories or industrial settings, ensuring that materials are used efficiently.

Specific Heat Capacity

Specific heat capacity is a property of a substance that tells us how much heat energy is required to raise the temperature of 1 gram of a substance by 1 degree Celsius (or 1 Kelvin). It is an intrinsic property, meaning it is characteristic of the substance and does not change with the amount.
In this context, the specific heat capacity helps determine how much heat is absorbed or released by the solution in the calorimeter:

• Specific heat capacity (c) = 4.2 J/g·K for the solutions.

This value is typically given or known for the solutions used, as it helps in calculating the heat changes during a reaction using the formula q = m × c × ΔT.
Understanding specific heat capacity is crucial in thermodynamics and heat transfer, as it influences the degree to which temperature changes when energy is added or removed from a system. It makes predicting temperature changes in chemical processes more accurate, allowing for better control and understanding of energy changes in reactions.