# Enthalpy

#### Prerequisites

When you've studied the energy changes associated with chemical reactions in your chemistry class, the quantity you've looked at was the change in enthalpy. But what is enthalpy really? Is it just the same thing as the change in energy? If not, what's the difference? And when is it useful to use enthalpy instead of energy?

The key to remember is that enthalpy, energy, and other energy-like quantities of thermodynamics are generally used to measure changes in the amount of energy of a system under particular conditions. To distinguish whether the change in the amount of energy in a system is called a change in internal energy or enthalpy or something else, we need to know which variables are held constant, and which variables are allowed to change. The main variables are temperature ($T$), pressure ($p$), volume ($V$), and the number of molecules ($N$).

## Constant-pressure processes are different from constant-volume processes

When a chemical reaction takes place, energy may be transformed from the chemical level (associated with the motion and interactions of electrons inside atoms) to the thermal level (associated with the motion and interactions of atoms and molecules). If this reaction takes place inside a sealed container with rigid walls (i.e. constant volume), that would be enough information to end the story, at least as far as the energy in the system was concerned. The energy of the set of molecules in our system would change from one form to another (say, chemical to thermal, for an exothermic reaction) and the total energy of the system would not change.

But frequently, in chemistry and biology, the systems we're interested in don't have constant volume, but instead have constant pressure since they often take place in the open and so are at atmospheric pressure.

## A simple change in the process conditions, from constant volume to constant pressure, affects the change in energy of the system!

Consider first a constant-volume example. Suppose you're boiling water on a stove in a closed pot. There are no holes in the lid for the steam to escape. The reaction H2O (l) → H2O (g) is endothermic: it takes energy to break the hydrogen bonds that hold the water molecules together in the liquid phase, so that they can move freely in the gas phase. As the water boils and becomes vapor, there are more molecules hitting the lid of the pot, and they are therefore exerting more force (per area) on the lid. Thus the pressure inside the pot is increasing. All the energy that goes into the pot from the burner is put into the molecules of water. Eventually the lid will blow off!

Now let's say, instead of a pot, you're boiling water inside a container that can freely expand and contract. Then the pressure inside this container will always end up equal to atmospheric pressure (i.e. the pressure outside): if the pressure is greater inside than outside, there will be a net force pushing the walls of the container outward, so it will expand (which decreases the pressure inside) until the pressure inside is equal to the pressure outside (so the net force is zero). If the pressure outside is greater than the pressure inside, the reverse will happen: the container will shrink (increasing the pressure) until the pressure is equal inside and outside.

So what happens when the water boils in the second case? Instead of the pressure building up inside the container, the container expands, keeping the pressure inside equal to atmospheric pressure. Thus, as the water (vapor) inside the container expands, it does work on its surroundings (since it's exerting a force over a displacement).

How much work do the water vapor do?

Work = (Force) * (displacement)
= (Force / area) * (area * displacement) = (Pressure) * (change in volume)

or in symbols, if a pressure $p$ pushes on an area of the wall, $A$, and moves it out by a small amount $Δx$, the change in volume is $ΔV = AΔx$ so the work done by the mass on the wall is

$$W = F\Delta x = \Bigg(\frac{F}{A}\bigg) (A\Delta x) = p\Delta V$$

So if we want to find the total energy that it takes to make this change happen at constant pressure, we have to consider not only the change in energy involved in breaking/making bonds for converting liquid to gas, we also have to add the energy to "make room" for the products of the reaction (i.e. the work involved in changing the volume).

This total energy is called the enthalpy. The symbol is $H$, and it is defined as $H = U + pV$ (internal energy + pressure * volume), for the reasons discussed above.

But what we typically care about is the change in enthalpy: $ΔH = ΔU + pΔV$. This is what you look up (for various reactions) in the back of your chemistry textbook. Since the change in enthalpy depends on pressure, the tables have to assume a pressure — typically 1 atmosphere (about 100 kPa).

Enthalpy is the relevant quantity for systems at constant pressure. Biologically relevant systems at constant pressure include:

• Cells, since the membrane can deform to maintain the cell at atmospheric pressure
• Anything open to the environment (imagine the pot of boiling water had the lid off)

A positive change in enthalpy means that heat is being input to the system (and remember, this is after you consider the work involved in keeping the system at constant pressure); in other words, the process is endothermic. A negative change in enthalpy means that heat is being released from the system to its surroundings (again, after considering the work to keep the system at constant pressure).

Ben Dreyfus 12/26/2011, Wolfgang Losert 1/25/2013

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