Problem 38 - General Chemistry: Principles and Modern Applications

Question

A coffee-cup calorimeter contains \(100.0 \mathrm{mL}\) of \(0.300 \mathrm{M}\) HCl at \(20.3^{\circ} \mathrm{C}\). When \(1.82 \mathrm{g} \mathrm{Zn}(\mathrm{s})\) is added, the temperature rises to \(30.5^{\circ} \mathrm{C}\). What is the heat of reaction per mol Zn? Make the same assumptions as in Example \(7-4,\) and also assume that there is no heat lost with the \(\mathrm{H}_{2}(\mathrm{g})\) that escapes. $$\mathrm{Zn}(\mathrm{s})+2 \mathrm{H}^{+}(\mathrm{aq}) \longrightarrow \mathrm{Zn}^{2+}(\mathrm{aq})+\mathrm{H}_{2}(\mathrm{g})$$

Step-by-Step Solution

Answer

The heat of reaction per mole of Zn is \(-35.3 \, kJ/mol\). Note that the exact value may differ slightly due to rounding during the calculations.

Step 1: Determine the heat absorbed

Start by using the formula q = mcΔT to find the heat absorbed by the water. Here, m is the mass of the water (which can be found by taking the volume in mL and assuming 1 mL of water is 1g so mass is 100g), c is the specific heat capacity of water (4.184 J/g°C), and ΔT is the change in temperature (30.5°C - 20.3°C).

Step 2: Convert heat absorbed to kJ

The previous calculation gave the heat absorbed in Joules. Because the problem asks for the answer in kJ/mol, convert Joules to kiloJoules by dividing by 1000. This will give the number of kJ absorbed by the reaction.

Step 3: Calculate the number of moles of zinc reacted

To find the heat per mole of zinc, it's necessary to first find out how many moles of zinc were used. We do this by dividing the mass of zinc given (1.82g) by the molar mass of zinc (65.38 g/mol).

Step 4: Determine the heat of reaction per mol of Zn

Finally, divide the heat absorbed (calculated in step 2) by the number of moles of zinc reacted (calculated in step 3) to find the heat of reaction per mole of zinc.

Step 5: Assign the proper sign to the heat of reaction

Since heat was released by the reaction (it was absorbed by the water causing temperature to rise), the heat of reaction is exothermic and therefore negative.

Key Concepts

Heat of Reaction

In calorimetry, the heat of reaction refers to the energy change that occurs during a chemical reaction. The focus is on the energy exchanged between the system (in this case, the reaction) and its surroundings. To determine the heat of reaction, we use the principle of conservation of energy, which tells us that energy cannot be created or destroyed, only transferred.

The procedure usually involves measuring the temperature change in a known quantity of water. We use the formula \( q = mc\Delta T \) to calculate the energy (heat, \( q \)) absorbed or released:

• \( m \) is the mass of the water, here equivalent to its volume (100g since \( 1 \text{ mL} = 1 \text{ g} \))
• \( c \) is the specific heat capacity of water, approximately \( 4.184 \text{ J/g°C} \)
• \( \Delta T \) is the temperature change (from \(20.3^{\circ} \text{C} \) to \(30.5^{\circ} \text{C} \))

These values help us calculate \( q \), the energy transferred, in Joules. Converting this to kilojoules gives us a more manageable number, aligning with reactions where we often speak in terms of \( \text{kJ/mol} \).

Since the reaction between zinc and hydrochloric acid causes the temperature to rise, it’s exothermic, meaning it releases heat. Thus, the heat of reaction will be a negative value.

Enthalpy Change

While similar to heat of reaction, enthalpy change is often discussed in broader terms in chemistry. It represents the total heat content of a system and changes as a function of pressure, volume, and temperature in most reactions. However, in a closed system like a calorimeter, the focus is primarily on temperature change.

For constant pressure processes, such as those assumed in many calorimeter exercises, we approximate enthalpy change and heat of reaction as identical. Thus, what we often calculate as the heat exchanged, \( q \), provides insight into the enthalpy change, \( \Delta H \).

When we say the "enthalpy change for the reaction is negative," we relate this directly to the exothermic nature of the reaction—energy flows from the reaction to heat the water. To determine \( \Delta H \) per mole of a reactant like zinc in this exercise, we divide the calculated energy change by the moles of zinc involved.

Enthalpy also considers other sorts of energy in systems, but in solutions like this exercise, focus primarily remains on heat, illustrating why calorimetry effectively models these reactions.

Zinc Hydrochloric Acid Reaction

This reaction is quintessential in understanding exothermic processes and their measurements. When zinc \((\text{Zn})\) reacts with hydrochloric acid \((\text{HCl})\), hydrogen gas \((\text{H}_2)\) is released, and zinc chloride \((\text{Zn}^{2+})\) is formed. This can be represented as:
\[ \text{Zn(s) + 2H}^{+}\text{(aq)} \rightarrow \text{Zn}^{2+}\text{(aq) + H}_2\text{(g)} \]

Typically, metals like zinc donate electrons to the hydrogen ions, allowing the release of hydrogen gas. This electron transfer liberates energy, seen as an increase in temperature when measured in a calorimeter.

To deeply understand this reaction, consider the oxidation-reduction nature involved:

• Zinc is oxidized from 0 to \(+2\) oxidation state.
• Protons from hydrochloric acid are reduced as they convert into hydrogen gas.

The heat released, which students measure under these controlled conditions, gives insight into the relative strengths of different substances in donating and accepting electrons. The transformation of chemical into thermal energy during this reaction is what drives the observed temperature rise in the calorimeter. This visualization of the zinc and hydrochloric acid reaction is crucial for beginners, as it directly ties the chemical equations we learn to the physical changes we can measure in class.