Question

Assume that these reactions occur under constant atmospheric pressure. What is the sign of \(w\) for each? (a) \(\mathrm{CaO}(\mathrm{s})+3 \mathrm{C}(\mathrm{s}) \longrightarrow \mathrm{CaC}_{2}(\mathrm{~s})+\mathrm{CO}(\mathrm{g})\) (b) \(2 \mathrm{C}_{6} \mathrm{H}_{6}(\ell)+15 \mathrm{O}_{2}(\mathrm{~g}) \longrightarrow 12 \mathrm{CO}_{2}(\mathrm{~g})+6 \mathrm{H}_{2} \mathrm{O}(\ell)\)

Short answer

(a) \( w \) is negative; (b) \( w \) is positive.

Step-by-step solution

Understanding the concept

In chemistry, when a reaction occurs under constant pressure, the work done by the system can be determined by examining the change in the number of moles of gas. The work done, denoted as \( w \), is related to the change in the volume of gases because work is done by a system when it expands or contracts against an external pressure. When more moles of gas molecules are produced than consumed, \( w \) is negative (the system does work on the surroundings and loses energy). If fewer moles are produced than consumed, \( w \) is positive (work is done on the system by the surroundings). If the moles of gas do not change, \( w \) is zero.

Analyzing reaction (a)

In reaction (a): \[ \mathrm{CaO}( ext{s}) + 3 \mathrm{C}( ext{s}) \longrightarrow \mathrm{CaC}_{2}( ext{s}) + \mathrm{CO}( ext{g}) \]- Reactants: 0 moles of gas (all are solids).- Products: 1 mole of gas (CO).The reaction produces 1 mole of gas, therefore more gas moles are produced than consumed. This means that the system expands; therefore, \( w \) is negative.

Analyzing reaction (b)

In reaction (b): \[ 2 \mathrm{C}_{6} \mathrm{H}_{6}(\ell)+15 \mathrm{O}_{2}(\text{g}) \longrightarrow 12 \mathrm{CO}_{2}(\text{g})+6 \mathrm{H}_{2} \mathrm{O}(\ell) \]- Reactants: 15 moles of gas (O2).- Products: 12 moles of gas (CO2).The reaction consumes more gas moles than it produces (15 moles to 12 moles), meaning there is a decrease in the number of gas moles and contraction, thus, \( w \) is positive.

Key concepts

Work in Chemical Reactions

In chemical reactions, particularly those involving gases, work can be done either by or on the system. This work, denoted by the symbol \( w \), relates to changes in volume due to gas expansion or compression. Under constant pressure conditions, which are common in many reactions, the concept of work becomes especially significant. For any reaction occurring at constant pressure, if the volume of gas increases, the system has done work on the surroundings because it pushes against the external pressure. This work is often considered energy leaving the system, thus \( w \) is negative. On the other hand, if the gas volume decreases, energy is put into the system (by the surroundings compressing it), making \( w \) positive. If there is no change in gas volume, no work is done, resulting in \( w \) being zero.Understanding how to evaluate these changes helps in solving problems related to energy changes in chemical reactions, especially when it comes to predicting the sign of \( w \). This evaluation is primarily focused on the change in moles of gas since solids and liquids generally have negligible volume changes.

Gas Volume Change

When a chemical reaction involves gases, the change in the number of moles of gas can lead to a change in volume. According to the ideal gas law, more moles of gas will occupy more volume under constant temperature and pressure conditions. Here's how to determine the effect of gas volume change:

• Calculate the total moles of gas on both the reactant and product side of the equation.
• If the moles of gas increase, the system expands, indicating a volume increase.
• If the moles of gas decrease, the system contracts, indicating a volume decrease.
• If the moles of gas remain the same, the volume does not change.

For example, in reaction (a) of our problem, 1 mole of gas is produced while none were consumed, signaling a volume increase. In contrast, reaction (b) results in fewer moles of gas, indicating a volume decrease. Knowing how to quantify these changes is crucial for evaluating the work associated with the reaction, especially when reactions have differing impacts on the energy exchange with the surroundings.

Constant Pressure Reactions

Chemistry often considers reactions at constant pressure, as conditions like atmospheric pressure are easy to maintain. Under constant pressure, the work done by or on the system is particularly tied to volume changes of gases. The relationship between pressure, volume, and work under constant conditions is given by the equation \( w = -P \Delta V \), where \( \Delta V \) is the change in volume, and \( P \) is the constant pressure. This equation helps clarify why expansions, where \( \Delta V \) is positive, result in negative \( w \), and why compressions, where \( \Delta V \) is negative, make \( w \) positive.In real-world applications, this notion assists in predicting energy changes. For constant pressure scenarios:

• An increase in gas moles (from reactants to products) usually signifies expansion and thus \( w \) is negative.
• A decrease denotes compression, and accordingly, \( w \) is positive.

In everyday chemical reactions, knowing whether a system is gaining or losing energy helps us design processes and understand natural phenomena that depend heavily on energy transformations.