Question

Consider the reaction $$ \begin{array}{r} 2 \mathrm{HCl}(a q)+\mathrm{Ba}(\mathrm{OH})_{2}(a q) \longrightarrow \mathrm{BaCl}_{2}(a q)+2 \mathrm{H}_{2} \mathrm{O}(l) \\ \Delta H=-118 \mathrm{~kJ} \end{array} $$ 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

The final temperature of the mixture when 100.0 mL of 0.500 M HCl is mixed with 300.0 mL of 0.100 M Ba(OH)₂ at an initial temperature of 25.0°C is 23.24°C.

Step-by-step solution

Calculate moles of reactants present

First, we'll determine the number of moles for each reactant in the solution using the given volumes and molarities: HCl moles = volume × molarity = 100 mL × 0.500 M = 50 mmol (0.0500 mol) Ba(OH)2 moles = volume × molarity = 300 mL × 0.100 M = 30 mmol (0.0300 mol)

Identify limiting reactant

Next, we'll find the limiting reactant by comparing the mole ratio of reactants to the stoichiometric coefficients from the balanced equation. Divide the moles of each reactant by their corresponding coefficients, and determine the smallest value: HCl: \(\frac{0.0500 \ mol}{2}\) = 0.0250 Ba(OH)₂: \(\frac{0.0300 \ mol}{1}\) = 0.0300 The smallest value is 0.0250, which corresponds to HCl. This means that HCl is the limiting reactant.

Calculate the heat produced by the reaction

Now that we know the limiting reactant, we can determine the heat produced by the reaction (q). We'll use the molar ratio between the limiting reactant and the balanced equation to find the moles of reaction, and then use the given ΔH to calculate the heat produced: Moles of reaction = \(\frac{0.0500 \ mol \ \text{HCl}}{2}\) = 0.0250 mol (due to the stoichiometric ratio, 2 HCl → 1 BaCl₂) Heat produced (q) = moles of reaction × ΔH = 0.0250 mol × (-118 kJ/mol) = -2.95 kJ The negative sign indicates that the reaction is exothermic and releases heat.

Calculate the final temperature of the mixture

Finally, we'll use the specific heat capacity formula to find the final temperature of the mixture: q = mcΔT Rearrange the formula to solve for ΔT: ΔT = \(\frac{q}{mc}\) = \(\frac{-2950 \ J}{(400.0 \ g)(4.18 \ J/^{\circ} C \cdot g)}\) = -1.76°C Now, we'll add the ΔT to the initial temperature (25.0°C) to find the final temperature of the mixture: Final temperature = Initial temperature + ΔT = 25.0°C - 1.76°C = 23.24°C The final temperature of the mixture is 23.24°C.

Key concepts

Understanding Stoichiometry

Stoichiometry is essentially the chemistry version of a recipe. It's a section of chemistry that involves using balanced chemical equations to calculate the relative quantities of reactants and products involved in a reaction. Think of it as the mathematical relationship between the elements and compounds in a chemical reaction.

Let's dive into an example. The balanced chemical equation for the reaction between hydrochloric acid (HCl) and barium hydroxide (Ba(OH)₂) is:
\[2 \mathrm{HCl}(aq) + \mathrm{Ba(OH)}_{2}(aq) \longrightarrow \mathrm{BaCl}_{2}(aq) + 2 \mathrm{H}_{2}\mathrm{O}(l)\]
This equation tells us that two moles of HCl react with one mole of Ba(OH)₂ to produce one mole of BaCl₂ and two moles of water. It provides the exact ratios needed to calculate how much product can be formed from a given amount of reactants, which is a fundamental aspect of stoichiometry.

Identifying the Limiting Reactant

In many chemical reactions, one reactant will run out before the others, and this reactant is known as the limiting reactant. It essentially sets the stage for how much product can be formed, much like having a limited amount of flour when you're trying to bake a big batch of cookies. The remaining reactants, which are not entirely consumed, are called excess reactants.

In our example problem, we identified the limiting reactant by calculating the number of moles of each reactant and comparing them to the stoichiometric coefficients in the balanced equation. The reactant that yields the smallest amount when divided by its coefficient is the limiting reactant because it will be completely consumed first, thus stopping the reaction. With the concept of limiting reactants, we're able to predict how a reaction will progress and determine the maximum amount of product that can be formed.

The Nature of Exothermic Reactions

Exothermic reactions are the cozy campfires of the chemical world – they release energy in the form of heat, causing the surrounding environment to warm up. This is why the temperature of a system often increases following an exothermic reaction, as the energy stored in chemical bonds is transformed into thermal energy. The \( \Delta H \) value in our example is a negative number, which is a telltale sign of an exothermic reaction.

When calculating the heat released in an exothermic reaction, remember that the amount of heat (q) is directly proportional to the number of moles of the limiting reactant that reacts. The negative value of \( \Delta H \) tells us the direction of heat flow, from the chemcials to the surroundings, hence the negative sign representing a release of energy.

Role of Specific Heat Capacity in Temperature Change

The specific heat capacity is a measure of how much energy is needed to raise the temperature of a certain mass of substance by one degree Celsius. It's the reason why some things (like water) take longer to heat up than others – every material has its unique 'thermal stubbornness' level.

In the given exercise, using the specific heat capacity allows us to determine the final temperature of the mixture after the reaction has occurred. It's a crucial factor in calculating temperature change because it helps us understand how the energy produced from a reaction (the heat) translates into a measurable change in temperature. By knowing the mass of our system, the heat released, and the specific heat capacity, we can solve for the temperature change (\( \Delta T \)) and hence, find the final temperature of the solution.