## Physical Chemistry Lesson of the Day – Standard Heats of Formation

The standard heat of formation, ΔHfº, of a chemical is the amount of heat absorbed or released from the formation of 1 mole of that chemical at 25 degrees Celsius and 1 bar from its elements in their standard states.  An element is in its standard state if it is in its most stable form and physical state (solid, liquid or gas) at 25 degrees Celsius and 1 bar.

For example, the standard heat of formation for carbon dioxide involves oxygen and carbon as the reactants.  Oxygen is most stable as O2 gas molecules, whereas carbon is most stable as solid graphite.  (Graphite is more stable than diamond under standard conditions.)

To phrase the definition in another way, the standard heat of formation is a special type of standard heat of reaction; the reaction is the formation of 1 mole of a chemical from its elements in their standard states under standard conditions.  The standard heat of formation is also called the standard enthalpy of formation (even though it really is a change in enthalpy).

By definition, the formation of an element from itself would yield no change in enthalpy, so the standard heat of reaction for all elements is zero.

## Physical Chemistry Lesson of the Day – Hess’s Law

Hess’s law states that the change in enthalpy of a multi-stage chemical reaction is just the sum of the changes of enthalpy of the individual stages.  Thus, if a chemical reaction can be written as a sum of multiple intermediate reactions, then its change in enthalpy can be easily calculated.  This is especially helpful for a reaction whose change in enthalpy is difficult to measure experimentally.

Hess’s law is a consequence of the fact that enthalpy is a state function; the path between the reactants and the products is irrelevant to the change in enthalpy – only the initial and final values matter.  Thus, if there is a path for which the intermediate values of $\Delta H$ are easy to obtain experimentally, then their sum equal the $\Delta H$ for the overall reaction.

## Physical Chemistry Lesson of the Day – State Functions vs. Path Functions

Today’s lesson may seem mundane; despite its subtlety, it is actually quite important.  I needed to spend some time to learn it and digest it, and it was time well spent – these concepts are essential for understanding much of thermodynamics.  For brevity, I have not dived into the detailed mathematics of exact differentials, though I highly recommend you to learn it and review the necessary calculus.

Some thermodynamic properties of a system can be described by state variables, while others can be described by path variables.

A state variable is a variable that depends only on the final and initial states of a system and not on the path connecting these states.  Internal energy and enthalpy are examples of state functions.  For example, in a previous post on the First Law of Thermodynamics, I defined the change in internal energy, $\Delta U$, as

$\Delta U = \int_{i}^{f} dU = U_f - U_i$.

State variables can be calculated by exact differentials.

A path variable is a variable that depends on the sequence of steps that takes the system from the initial state to the final state.  This sequence of steps is called the path.  Heat and work are examples of path variables.  Path variables cannot be calculated by exact differentials.  In fact, the following quantities may seem to have plausible interpretations, but they actually do not exist:

• change in heat ($\Delta q$)
• initial heat ($q_i$)
• final heat ($q_f$)
• change in work ($\Delta w$)
• initial work ($w_i$)
• final work ($w_f$)

There is no such thing as heat or work being possessed by a system.  Heat and work can be transferred between the system and the surroundings, but the end result is an increase or decrease in internal energy; neither the system or the surroundings possesses heat or work.

A state/path variable is also often called a state/path function or a state/path quantity.

## Physical Chemistry Lesson of the Day – Standard Heats of Reaction

The change in enthalpy of a chemical reaction indicates how much heat is absorbed or released by the system.  This is valuable information in chemistry, because the exchange in heat affects the reaction conditions and the surroundings, and that needs to be managed and taken into account – in theory, in the laboratory, in industry or in nature in general.

Chemists often want to compare the changes in enthalpy between different reactions.  Since changes in enthalpy depend on both temperature and pressure, we need to control for these 2 confounding variables by using a reference set of temperature and pressure.  This set of conditions is called the standard conditions, and it sets the standard temperature at 298 degrees Kelvin and the standard pressure at 1 bar.  (IUPAC changed the definition of standard pressure from 1 atmosphere to 1 bar in 1982.  The actual difference in pressure between these 2 definitions is very small.)

The standard enthalpy of reaction (or standard heat of reaction) is the change in enthalpy of a chemical reaction under standard conditions; the actual number of moles are specified by the coefficients of the balanced chemical equation.  (Since enthalpy is an extensive property, the same reaction under standard conditions could have different changes in enthalpy with different amounts of the reactants and products.  Thus, the number of moles of the reaction must be standardized somehow when defining the standard enthalpy of reaction.)  The standard enthalpy of reaction has the symbol ΔHº; the º symbol indicates the standard conditions.

## Physical Chemistry Lesson of the Day – The Effect of Temperature on Changes in Internal Energy and Enthalpy

When the temperature of a system increases, the kinetic and potential energies of the atoms and molecules in the system increase.  Thus, the internal energy of the system increases, which means that the enthalpy of the system increases – this is true under constant pressure or constant volume.

Recall that the heat capacity of a system is the amount of energy that is required to raise the system’s temperature by 1 degree Kelvin.  Since the heat absorbed by the system in a thermodynamic process is the increase in enthalpy of the system, the heat capacity is just the change in enthalpy divided by the change in temperature.

$C = \Delta H \div \Delta T$.

## Physical Chemistry Lesson of the Day – The Difference Between Changes in Enthalpy and Changes in Internal Energy

Let’s examine the difference between a change enthalpy and a change in internal energy.  It helps to think of the following 2 scenarios.

• If the chemical reaction releases a gas but occurs at constant volume, then there is no pressure-volume work.  The only way for energy to be transferred between the system and the surroundings is through heat.  An example of a system under constant volume is a bomb calorimeter.  In this case,

$\Delta H = \Delta U + P \Delta V = \Delta U + 0 = q - w + 0 = q - 0 + 0 = q$

This heat is denoted as $q_v$ to indicate that this is heat transferred under constant volume.  In this case, the change in enthalpy is the same as the change in internal energy.

• If the chemical reaction releases a gas and occurs at constant pressure, then energy can be transferred between the system and the surroundings through heat and/or work.  Thus,

$\Delta H = \Delta U + P \Delta V = q - w + P \Delta V = q$

This heat is denoted as $q_p$ to indicate that this is heat transferred under constant pressure.  Thus, as the gas forms inside the cylinder, the piston pushes against the constant pressure that the atmosphere exerts on it.  The total energy released by the chemical reaction allows some energy to be used for the pressure-volume work, with the remaining energy being released via heat.  (Recall that these are the 2 ways for internal energy to be changed according to the First Law of Thermodynamics.)  Thus, the difference between enthalpy and internal energy arises under constant pressure – the difference is the pressure-volume work.

Reactions under constant pressure are often illustrated by a reaction that releases a gas in cylinder with a movable piston, but they are actually quite common.  In fact, in chemistry, reactions under constant pressure are much more common than reactions under constant volume.  Chemical reactions often happen in beakers, flasks or any container open to the constant pressure of the atmosphere.

## Physical Chemistry Lesson of the Day – Enthalpy

The enthalpy of a system is the system’s internal energy plus the product of the pressure and the volume of the system.

$H = U + PV$.

Just like internal energy, the enthalpy of a system cannot be measured, but a change in enthalpy can be measured.  Suppose that the only type of work that can be performed on the system is pressure-volume work; this is a realistic assumption in many chemical reactions that occur in a beaker, a flask, or any container that is open to the constant pressure of the atmosphere.  Then, the change in enthalpy of a system is the change in internal energy plus the pressure-volume work done on the system.

$\Delta H = \Delta U + P\Delta V$.