Question

In a coffee-cup calorimeter, \(150.0 \mathrm{~mL}\) of \(0.50 \mathrm{M} \mathrm{HCl}\) is added to \(50.0 \mathrm{~mL}\) of \(1.00 \mathrm{M} \mathrm{NaOH}\) to make \(200.0 \mathrm{~g}\) solution at an initial temperature of \(48.2^{\circ} \mathrm{C}\). If the enthalpy of neutralization for the reaction between a strong acid and a strong base is \(-56 \mathrm{~kJ} / \mathrm{mol}\), calculate the final temperature of the calorimeter contents. Assume the specific heat capacity of the solution is \(4.184 \mathrm{~J} / \mathrm{g} \cdot{ }^{\circ} \mathrm{C}\) and assume no heat loss to the surroundings.

Short answer

The final temperature of the calorimeter contents after adding \(0.50 \mathrm{M}\) HCl to \(1.00 \mathrm{M}\) NaOH is \(45.4^{\circ} \mathrm{C}\).

Step-by-step solution

Calculate the moles of HCl and NaOH

Given the volumes and concentrations of the HCl and NaOH solutions, we first need to determine the moles of each reactant. For the HCl solution: Volume, V = 150 mL = 0.150 L Concentration, C = 0.50 M Moles HCl, n = C × V = 0.50 mol/L × 0.150 L = 0.075 mol For the NaOH solution: Volume, V = 50 mL = 0.050 L Concentration, C = 1.00 M Moles NaOH, n = C × V = 1.00 mol/L × 0.050 L = 0.050 mol

Determine the limiting reactant

Since the neutralization reaction between HCl and NaOH has a 1:1 stoichiometric ratio, the reactant with fewer moles will be the limiting factor. In this case, NaOH has fewer moles (0.050 mol) than HCl (0.075 mol), so NaOH will be the limiting reactant.

Calculate the heat evolved from the neutralization reaction

Given the enthalpy of neutralization for the reaction, ΔH_neutralization = -56 kJ/mol, we can calculate the heat evolved during the reaction, Q, using the following equation: Q = moles of limiting reactant × ΔH_neutralization Q = 0.050 mol × (-56 kJ/mol) Q = -2.8 kJ The heat evolved during the reaction is -2.8 kJ, which means that the reaction is exothermic and 2.8 kJ of heat is released to the solution.

Determine the final temperature of the calorimeter contents

Now, using the heat capacity equation Q = mcΔT (where m is the mass, c is the specific heat capacity, and ΔT is the change in temperature), we can determine the final temperature of the calorimeter contents: Q = mcΔT The mass of the solution, m = 200 g The specific heat capacity, c = 4.184 J/g °C The initial temperature, T_initial = 48.2 °C The heat evolved, Q = -2.8 kJ = -2800 J (converting kilojoules to joules) Solving for the change in temperature, ΔT: -2800 J = (200 g)(4.184 J/g °C)(ΔT) ΔT = -2800 J / (200 g × 4.184 J/g °C) = -2.8 °C Now, we can find the final temperature of the calorimeter contents: T_final = T_initial + ΔT T_final = 48.2 °C - 2.8 °C T_final = 45.4 °C The final temperature of the calorimeter contents is 45.4 °C.

Key concepts

Calorimetry

Calorimetry is a technique used to measure the amount of heat involved in a chemical or physical process. In the context of the enthalpy of neutralization, a coffee-cup calorimeter can measure the heat change when an acid reacts with a base. This device is typically a well-insulated container that helps to ensure minimal heat loss to the external environment.

During a calorimetry experiment, data like the initial temperature of the solution, the mass of the solution, and the temperature change upon reaction are collected. These parameters are crucial for calculating the heat transfer using the formula \( Q = mc\Delta T \) where \( Q \) is the heat absorbed or released, \( m \) is the mass of the solution, \( c \) is the specific heat capacity, and \( \Delta T \) represents the change in temperature.

Stoichiometry

Stoichiometry is the mathematical relationship between the quantities of reactants and products in a chemical reaction. It's based on the balanced chemical equations that represent these reactions. For the neutralization reaction between \( \text{HCl} \) and \( \text{NaOH} \), stoichiometry dictates a 1:1 molar ratio, meaning one mole of \( \text{HCl} \) reacts with one mole of \( \text{NaOH} \) to produce water and salt.

In a calorimetry experiment, to determine the heat change, you need to know the moles of the limiting reactant. This is found using stoichiometry by comparing the mole ratios of reactants. From the given concentrations and volumes of \( \text{HCl} \) and \( \text{NaOH} \) in the exercise, the limiting reactant is identified, which then allows for the calculation of heat evolved or absorbed.

Exothermic Reactions

Exothermic reactions are chemical reactions that release energy in the form of heat. This concept is at the heart of understanding why the temperature in our calorimetry example increases. The enthalpy of neutralization for strong acids and bases is typically negative, indicating an exothermic reaction.

When \( \text{HCl} \) reacts with \( \text{NaOH} \), the reaction releases heat into the surrounding solution, causing the temperature to rise. The heat energy evolved can be calculated by multiplying the moles of the limiting reactant by the reaction's enthalpy change (\( Q = n\Delta H_{neutralization} \)). This calculated value for \( Q \) is then used in conjunction with the specific heat capacity to determine the final temperature after the exothermic process has occurred.

Specific Heat Capacity

Specific heat capacity is the amount of heat energy required to raise the temperature of one gram of a substance by one degree Celsius. In our exercise, the specific heat capacity of the solution is used to determine the final temperature after the neutralization reaction occurs.

The formula \( Q = mc\Delta T \) includes the specific heat capacity (\( c \) variable), and when rearranged to solve for the change in temperature (\( \Delta T \) variable), it highlights how this property of the material impacts the overall energy dynamics of the system. Essentially, the specific heat capacity tells us how resistant the solution is to changing temperature when it absorbs or releases heat. In the context of the calorimetry exercise, knowing the specific heat capacity of the solution allows for precise calculations of temperature changes during the neutralization.