Question

Under what circumstances are \(\Delta E\) and \(\Delta H\) essentially equal?

Short answer

ΔE and ΔH are essentially equal when there are no gas mole changes or minimal volume changes.

Step-by-step solution

Understand the Relationship between ΔE and ΔH

The change in internal energy, \(\Delta E\), and the change in enthalpy, \(\Delta H\), are related through the equation \(\Delta H = \Delta E + \Delta (PV)\), where \(P\) is pressure and \(V\) is volume. Understanding this relationship is crucial for identifying when these two quantities are equal.

Examine the Ideal Gas Behavior

For ideal gases, the equation becomes \(\Delta H = \Delta E + \Delta nRT\), where \(\Delta n\) is the change in moles of gas, \(R\) is the ideal gas constant, and \(T\) is the temperature. When there is no change in the number of moles of gas (\(\Delta n = 0\)), the equations simplify to \(\Delta H = \Delta E\). This typically happens in reactions where the number of moles of gaseous products equals the moles of gaseous reactants.

Consider Volume and Pressure Changes

When the reaction occurs with negligible changes in volume or at constant volume (so that \(\Delta V = 0\)), the \(PV\) term in the enthalpy equation impacts less significantly on \(\Delta H\). This circumstance makes \(\Delta E\) and \(\Delta H\) nearly identical.

Account for Solids and Liquids

In reactions involving only solids and liquids, changes in volume are generally negligible compared to the gaseous phase, thus making the \(PV\) work term minimal. Under such conditions, \(\Delta E\) and \(\Delta H\) can be assumed nearly the same.

Key concepts

Internal Energy

Internal energy, symbolized as \(E\), is a fundamental concept in thermodynamics that represents the total energy contained within a system. It includes both the kinetic and potential energy of the particles that make up the system. The change in internal energy, denoted \(\Delta E\), results from heat transfer or work done on or by the system.
In chemical reactions, \(\Delta E\) is crucial as it reflects the energy absorbed or released during the process. It's important for students to understand that this internal energy change is an intrinsic property of the system and is independent of how the process is carried out.
A key feature of internal energy is that it accounts for all forms of energy but does not consider the effects of pressure and volume explicitly. This sets the stage for comparing it with enthalpy (\(H\)), which is another important thermodynamic quantity related to heat changes under constant pressure conditions.

Enthalpy

Enthalpy, denoted as \(H\), is another critical concept in thermodynamics that encompasses both the internal energy of a system and the energy associated with the system’s pressure and volume. It is given by the equation \(H = E + PV\).
This equation highlights that enthalpy includes the internal energy as well as the product of pressure \(P\) and volume \(V\). Thus, enthalpy can be thought of as a measure of the total heat content in a system.
In reactions occurring at constant pressure, the change in enthalpy \(\Delta H\) reflects the heat absorbed or released, making it particularly useful for studying chemical reactions and phase changes.
When pressure-volume work is insignificant, such as in reactions involving only solids and liquids or when there is no change in the number of moles of gas, we have \(\Delta H \approx \Delta E\). This simplification is essential in understanding reactions under specific conditions.

Ideal Gas Behavior

Ideal gas behavior is a simplifying assumption used in thermodynamics to describe how gases behave under certain conditions. An ideal gas is one where the interactions between molecules are negligible and the volume of the molecules themselves is very small compared to the volume the gas occupies.
For ideal gases, the change in enthalpy and internal energy relationship is simplified by the equation \(\Delta H = \Delta E + \Delta nRT\), where \(\Delta n\) is the change in moles of gas, \(R\) is the ideal gas constant, and \(T\) is the temperature.
When there is no change in the number of moles of gas (\(\Delta n = 0\)), the equation reduces to \(\Delta H = \Delta E\). This equality typically occurs in reactions where the reactants and products have the same total number of gas moles.
Understanding ideal gas behavior is vital for students as it provides a foundation for predicting how gases will react and transform under various conditions of temperature and pressure.

Pressure and Volume Changes

Pressure and volume changes can significantly impact how we calculate changes in internal energy and enthalpy. The term \(\Delta (PV)\) in the enthalpy equation \(\Delta H = \Delta E + \Delta (PV)\) represents the work done due to changes in pressure \(P\) and volume \(V\).
A crucial scenario is when reactions occur at negligible volume change or constant volume, which implies \(\Delta V = 0\). In such cases, the term \(\Delta (PV)\) becomes minimal or zero, leading to \(\Delta H \approx \Delta E\).
For reactions involving only solids and liquids, volume changes are often so small that the pressure-volume work is not significant, aligning \(\Delta H\) closely with \(\Delta E\).
This understanding is important as it helps simplify calculations and interpretations of thermodynamic quantities in specific scenarios, especially in closed systems where external pressure is constant.