# 5.3: Enthalpy

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- 21723

Both heat and work represent energy transfer mechanisms. As discussed previously, the term "heat" refers to a *process *in which a body (the contents of a tea kettle, for example) acquires or loses energy as a direct consequence of its having a *different temperature *than its surroundings. Similarly, work refers to the transfer of energy that does not involve temperature differences. Hence, work, like energy, can take various forms, the most familiar being mechanical and electrical.

*Mechanical work*arises when an object moves a distance \(Δx\) against an opposing force \(f\): \[ w_{mechanical} = f Δx\]*Electrical work*is done when a body having a charge \(q\) moves through a potential difference \(ΔV\). \[w_{electrical}=q\Delta V\]

The unit of work is Joules. Work, like heat, exists only when energy is being transferred. When two bodies are placed in thermal contact and energy flows from the warmer body to the cooler one, this process is called “heat”. A transfer of energy to or from a system by any means other than heat is called “work”.

## Enthalpy as a Composite Function

To further understand the relationship between heat flow (q) and the resulting change in internal energy (\(ΔU\)), we can look at two sets of limiting conditions: reactions that occur at constant volume and reactions that occur at constant pressure. We will assume that PV work is the only kind of work possible for the system, so we can substitute its definition from Equation \(\ref{18.5}\) into Equation \(\ref{5.2.5}\) to obtain the following:

\[ΔU = q − PΔV \label{5.2.5}\]

where the subscripts have been deleted.

If the reaction occurs in a closed vessel, the volume of the system is fixed, and ΔV is zero. Under these conditions, the heat flow (often given the symbol q_{v} to indicate constant volume) must equal ΔU:

At constant volume, no \(PV\) work can be done, and the change in the internal energy of the system is equal to the amount of heat transferred from the system to the surroundings or vice versa.

Many chemical reactions are not, however, carried out in sealed containers at constant volume but in open containers at a more or less constant pressure of about 1 atm. The heat flow under these conditions is given the symbol \(q_p\) to indicate constant pressure. Replacing q in Equation \(\ref{5.3.3}\) by \(q_p\) and rearranging to solve for q_{p},

Thus, at constant pressure, the heat flow for any process is equal to the change in the internal energy of the system plus the PV work done.

Because conditions of constant pressure are so important in chemistry, a new state function called enthalpy (H) is defined as

\[H =U + PV\]

At constant pressure, the change in the enthalpy of a system is as follows:

\[ΔH = ΔU + Δ(PV) = ΔU + PΔV \label{5.3.5}\]

Comparing the previous two equations shows that at constant pressure, the change in the enthalpy of a system is equal to the heat flow: \(ΔH = q_p\). This expression is consistent with our definition of enthalpy, where we stated that enthalpy is the heat absorbed or produced during any process that occurs at constant pressure.

At constant pressure, the change in the enthalpy of a system is equal to the heat flow: \(ΔH = q_p\).

## The Relationship between ΔH and ΔU

If ΔH for a reaction is known, we can use the change in the enthalpy of the system (Equation \(\ref{5.3.5}\)) to calculate its change in internal energy. When a reaction involves only solids, liquids, liquid solutions, or any combination of these, the volume does not change appreciably (ΔV = 0). Under these conditions, we can simplify Equation \(\ref{5.3.5}\) to ΔH = ΔU. If gases are involved, however, ΔH and ΔU can differ significantly. We can calculate ΔU from the measured value of ΔH by using the right side of Equation \(\ref{5.3.5}\) together with the ideal gas law, PV = nRT. Recognizing that Δ(PV) = Δ(nRT), we can rewrite Equation \(\ref{5.3.5}\) as follows:

\[ΔH = ΔU + Δ(PV) = ΔU + Δ(nRT) \label{5.3.6}\]

At constant temperature, Δ(nRT) = RTΔn, where Δn is the difference between the final and initial numbers of moles of gas. Thus

\[ΔU = ΔH − RTΔn \label{5.3.7}\]

For reactions that result in a net production of gas, Δn > 0, so ΔU < ΔH. Conversely, endothermic reactions (ΔH > 0) that result in a net consumption of gas have Δn < 0 and ΔU > ΔH. The relationship between ΔH and ΔU for systems involving gases is illustrated in Example \(\PageIndex{2}\).

As Example \(\PageIndex{2}\) illustrates, the magnitudes of ΔH and ΔU for reactions that involve gases are generally rather similar, even when there is a net production or consumption of gases.

## Summary

Enthalpy is a state function, and the change in enthalpy of a system is equal to the sum of the change in the internal energy of the system and the PV work done. Enthalpy is a state function whose change indicates the amount of heat transferred from a system to its surroundings or vice versa, at constant pressure. The change in the internal energy of a system is the sum of the heat transferred and the work done. At constant pressure, heat flow (q) and internal energy (U) are related to the system’s enthalpy (H). The heat flow is equal to the change in the internal energy of the system plus the PV work done. When the volume of a system is constant, changes in its internal energy can be calculated by substituting the ideal gas law into the equation for ΔU.