Question

For each process, state whether \(\Delta H\) is less than (more negative), equal to, or greater than \(\Delta E\) of the system. Explain. (a) An ideal gas is cooled at constant pressure. (b) A gas mixture reacts exothermically at fixed volume. (c) A solid reacts exothermically to yield a mixture of gases in a container of variable volume.

Short answer

(a) \(\textbacklash\textDelta H < \textbacklash\textDelta E\) \(b) \(\textbacklass\textDelta H = \textbacklash\textDelta E\) \(c) \(\textblash\textDelta H > \textdownlash\textDelta E\)

Step-by-step solution

Understanding Enthalpy \(\textbackslash\textDelta H\) and Internal Energy \(\textbackslash\textDelta E\)

Enthalpy \(\textbackslash\textDelta H\) is the heat change at constant pressure, while internal energy \(\textbackslash\textDelta E\) is the total energy change of the system. They are related by \(\textbackslash\textDelta H = \textbackslash\textDelta E + P\textbackslash\textDelta V\), where \(\textbackslash\textDelta V\) is the change in volume and \(\textbackslash P\) is the pressure.

Analyze cooling of an ideal gas at constant pressure

When an ideal gas is cooled at constant pressure, its volume decreases as temperature drops (according to the ideal gas law). Thus, \(\textbackslash\textDelta V < 0\). Since \(\textbackslash\textDelta H = \textbackslash\textDelta E + P\textbackslash\textDelta V\), and \(\textbackslash\textDelta V < 0\), it implies that \(\textbackslash\textDelta H < \textbackslash\textDelta E\). Therefore, enthalpy \(\textbackslash\textDelta H\) is less than internal energy \(\textbackslash\textDelta E\).

Analyze exothermic reaction of gas mixture at fixed volume

For an exothermic reaction at constant volume, there is no change in volume (\(\textbackslash\textDelta V = 0\)). Thus, \(\textbackslash\textDelta H = \textbackslash\textDelta E + P(0)\), implying \(\textbacklash\textDelta H = \textbackslash\textDelta E\). Therefore, enthalpy \(\textbacklash\textDelta H\) is equal to internal energy \(\textbacklash\textDelta E\).

Analyze exothermic reaction yielding gases in a variable volume container

When a solid reacts exothermically to produce gases in a container of variable volume, the volume of the system increases (\(\textbackslash\textDelta V > 0\)). Using \(\textbacklash\textDelta H = \textbacklash\textDelta E + P\textbackslash\textDelta V\), and considering \(\textbacklash\textDelta V > 0\), it follows that \(\textbacklash\textDelta H > \textbacklash\textDelta E\). Therefore, enthalpy \(\textbacklash\textDelta H\) is greater than internal energy \(\textbacklash\textDelta E\).

Key concepts

Enthalpy Change (ΔH)

Enthalpy change, or \(\textbackslash\textDelta H\), represents the heat absorbed or released by a system at constant pressure. It is a crucial concept in thermodynamics because it helps us understand heat transfer during various processes. For instance, when we heat water at atmospheric pressure, the energy we impart to the system is reflected in the change in enthalpy. Importantly, \(\textbackslash\textDelta H\) takes into account not just internal energy changes but also the work done by the system during volume changes. This makes it particularly useful in practical applications, like chemical reactions, where pressure is often kept constant.

Internal Energy (ΔE)

Internal energy, represented as \(\textbackslash\textDelta E\), is the total energy contained within a system. This includes all kinetic and potential energy contributions from molecular movements and interactions. Unlike enthalpy, internal energy is not directly concerned with the external pressure or volume changes. It's a measure of energy within the system's boundaries. Therefore, it is a more foundational concept that applies across different conditions. For example, when gas molecules collide or when intermolecular forces change, these are reflected in the internal energy.

Ideal Gas Law

The ideal gas law, written as \(PV = nRT\), connects pressure (P), volume (V), the amount of gas (n), the ideal gas constant (R), and temperature (T). This fundamental equation describes the behavior of ideal gases, providing insights into how gases respond to changes in pressure, volume, and temperature. For instance, in an ideal gas undergoing cooling at constant pressure, volume decreases as the temperature drops. This relationship helps us understand why \(\textbackslash\textDelta V \) becomes negative during such a process and influences the interplay between \(\textbackslash\textDelta H\) and \(\textbackslash\textDelta E\).

Exothermic Reactions

Exothermic reactions are chemical reactions that release energy in the form of heat. These reactions have a negative enthalpy change \((\textbackslash\textDelta H < 0)\) because the energy required to break bonds in the reactants is less than the energy released when new bonds form in the products. Examples include combustion reactions and many oxidation processes. In the context of internal energy \((\textbackslash\textDelta E)\), exothermic reactions at fixed volume conditions imply no work is done on or by the system. Thus, \(\textbackslash\textDelta H \) and \(\textbacklash\textDelta E\) are equal in such cases.

Volume Change (ΔV)

Volume change, denoted as \(\textbacklash\textDelta V\), is a crucial factor in thermodynamic processes. When the volume of a gas changes, work is done by or on the system, contributing to the energy change. For example, an expanding gas (positive \(\textbacklash\textDelta V\)) does work on its surroundings, which is reflected in the enthalpy change. This concept is vital in understanding processes like gas expansion due to temperature increase or pressure decrease. When evaluating processes, knowing whether volume increases or decreases helps predict whether \(\textbackslash\textDelta H \) will be greater than or less than \(\textbacklash\textDelta E\).