Question

Consider the reaction \(2 \mathrm{HCl}(a q)+\mathrm{Ba}(\mathrm{OH})_{2}(a q) \longrightarrow \mathrm{BaCl}_{2}(a q)+2 \mathrm{H}_{2} \mathrm{O}(l)\) \(\Delta H=-118 \mathrm{~kJ}\) Calculate the heat when \(100.0 \mathrm{~mL}\) of \(0.500 \mathrm{M} \mathrm{HCl}\) is mixed with \(300.0 \mathrm{~mL}\) of \(0.100 \mathrm{M} \mathrm{Ba}(\mathrm{OH})_{2}\). Assuming that the temperature of both solutions was initially \(25.0^{\circ} \mathrm{C}\) and that the final mixture has a mass of \(400.0 \mathrm{~g}\) and a specific heat capacity of \(4.18 \mathrm{~J} /{ }^{\circ} \mathrm{C} \cdot \mathrm{g}\), calculate the final temperature of the mixture.

Short answer

In this problem, HCl is the limiting reactant. By using the enthalpy change (-118 kJ), we calculate the heat generated by the reaction (-2950 J) and then determine the change in temperature (1.765$$^{\circ}$$C). This allows us to find the final temperature of the mixture which is approximately 26.8$$^{\circ}$$C.

Step-by-step solution

Determine the limiting reactant

Let's find the initial moles of both reactants: 1. Moles of HCl = Volume (L) × Molarity = 0.100 L × 0.500 M = 0.050 mol 2. Moles of Ba(OH)₂ = Volume (L) × Molarity = 0.300 L × 0.100 M = 0.030 mol Now, let's check the reaction's stoichiometry to find out which reactant will run out first. The reaction is given as: \[2 \mathrm{HCl} + \mathrm{Ba(OH)}_{2} \longrightarrow \mathrm{BaCl}_{2} + 2 \mathrm{H}_{2} \mathrm{O}\] This indicates that 2 moles of HCl react with 1 mole of Ba(OH)₂. So, we will divide the moles of each reactant by their respective stoichiometric coefficients and find the limiting reactant: 1. Moles of HCl divided by its coefficient = 0.050 mol / 2 = 0.025 2. Moles of Ba(OH)₂ divided by its coefficient = 0.030 mol / 1 = 0.030 Since 0.025 < 0.030, HCl is the limiting reactant.

Calculate the moles of the limiting reactant involved in the reaction

As HCl is the limiting reactant, we must use its moles in the reaction. Moles of HCl involved in the reaction are 0.050 moles.

Calculate the heat generated by the reaction

To calculate the heat generated by the reaction, we must use the enthalpy change of the reaction, which is -118 kJ. Since this enthalpy change corresponds to the given reaction's stoichiometry, we must first find the moles of HCl that correspond to the enthalpy change: \(\frac{\Delta H}{2 \, \text{moles }\mathrm{HCl}} = \frac{-118 \,\text{kJ}}{2} = -59 \,\mathrm{kJ/mol}\) Now, we use the moles of HCl involved in the reaction to find the total heat generated: Heat generated by the reaction = moles of HCl × -59 kJ/mol = 0.050 mol × -59 kJ/mol = -2.95 kJ = -2950 J

Calculate the change in temperature using the heat generated

We are given that the mass of the final mixture is 400.0 g and its specific heat capacity is 4.18 J/($$^{\circ}\)C·g). Using the heat generated by the reaction (-2950 J), we can calculate the change in temperature of the mixture using the following formula: \[\Delta T=\frac{-q}{mc}\] Where q is the heat generated, m is the mass of the mixture, and c is its specific heat capacity: \[\Delta T=\frac{-(-2950 \, \text{J})}{400.0 \, \text{g} \times 4.18 \,\mathrm{J}/^{\circ}\mathrm{C} \cdot \mathrm{g}} = \frac{2950}{1672}= 1.765 \,\,^{\circ}\mathrm{C}\]

Find the final temperature of the mixture

Lastly, to find the final temperature of the mixture, we will add the change in temperature to the initial temperature, which is 25.0$$^{\circ}$$C: Final temperature = Initial temperature + Change in temperature = 25.0$$^{\circ}$$C + 1.765$$^{\circ}$$C = 26.765$$^{\circ}$$C Thus, the final temperature of the mixture is approximately 26.8$$^{\circ}$$C.

Key concepts

Enthalpy Change

Enthalpy change is a crucial concept in thermochemistry, reflecting the heat absorbed or released during a chemical reaction under constant pressure. In the context of our reaction, the enthalpy change (\(\Delta H\)) is given as \(-118 \, \text{kJ}\). This negative sign indicates that the reaction is exothermic, meaning it releases heat to its surroundings.

Understanding enthalpy change allows scientists to predict whether a reaction requires or releases energy:

• A negative \(\Delta H\) indicates an exothermic reaction (heat is released).
• A positive \(\Delta H\) suggests an endothermic reaction (heat is absorbed).

For our given reaction, \(2 \,\text{HCl} + \text{Ba(OH)}_{2} \rightarrow \text{BaCl}_{2} + 2\, \text{H}_2\text{O}\), the enthalpy change per mole of HCl in the stoichiometric equation is determined by dividing the total \(\Delta H\) by the stoichiometric amount of HCl, aiding in calculating the heat exchanged per mole in practical scenarios. These calculations are essential in various fields, like chemical engineering, to assess the energy efficiency and safety of chemical processes.

Limiting Reactant

The concept of the limiting reactant is pivotal in stoichiometry as it determines how much product can be formed in a chemical reaction. In our reaction scenario, we identify the limiting reactant by calculating and comparing the moles of \(\text{HCl}\) and \(\text{Ba(OH)}_{2}\) before the reaction takes place.

Here's the general idea:

• The limiting reactant is the one that will be completely consumed first, thus limiting the extent of the reaction.
• Calculations involve comparing the actual mole ratios to the stoichiometric ratios in the balanced chemical equation.

For the given reaction, \(\text{HCl}\) is the limiting reactant because its mole to coefficient division yields the smallest number. This finding sets the stage for determining the amount of product produced and the heat (enthalpy change) it incurs. Identifying the limiting reactant is crucial in practical applications like yield optimization and cost efficiency in manufacturing.

Stoichiometry

Stoichiometry is the branch of chemistry that deals with the quantitative relationships between reactants and products in a chemical reaction. It is highly instrumental in predicting reaction yields and understanding reaction mechanisms.

Important aspects of stoichiometry include:

• Conversion of quantities of reactants to products using their balanced chemical equations.
• Understanding that stoichiometric coefficients indicate the ratios of moles of reactants and products involved.

In our reaction example, the balanced equation \(2 \,\text{HCl} + \text{Ba(OH)}_{2} \rightarrow \text{BaCl}_{2} + 2\, \text{H}_2\text{O}\) informs us that 2 moles of HCl react with 1 mole of \(\text{Ba(OH)}_{2}\) to produce 1 mole of \(\text{BaCl}_{2}\) and 2 moles of water. This allows us to precisely calculate the required reactants and expected products, essential in both lab and industry settings.

Specific Heat Capacity

Specific heat capacity is a property that describes how much heat is needed to change the temperature of a given mass by 1 degree Celsius. In this context, the specific heat capacity helps us determine how much the temperature of the reaction mixture will change when heat is absorbed or released.

Key aspects include:

• The specific heat capacity is given as \(4.18 \, \text{J}/^{\circ}\text{C} \cdot \text{g}\) for the mixture.
• Allows calculation of temperature change (\(\Delta T\)) using the formula \(\Delta T = \frac{-q}{mc}\) where \(q\) is the heat absorbed or released, \(m\) is the mass, and \(c\) is the specific heat capacity.

In our scenario, the heat generated by the reaction is transferred to the mixture, changing its temperature. The final temperature can be easily calculated by adding \(\Delta T\) to the initial temperature. Understanding this concept is instrumental in any situation where temperature regulation is critical, like in chemical reactions, climate studies, and engineering applications.