Introduction to Thermochemistry and Enthalpy

The First Law of Thermodynamics is essentially the application of the Law of Conservation of Energy to these systems. It dictates that any change in energy within a system must be balanced by an equal and opposite change in the surroundings:

$\Delta \mathrm{Energy}_{\text{universe}} = 0 = \Delta \mathrm{Energy}_{\text{system}} + \Delta \mathrm{Energy}_{\text{surroundings}}$

Internal Energy and Work

The internal energy ($\Delta E$) of a system can change through the exchange of heat ($q$) or work ($w$):

$\Delta E = q + w$

Energy exchange is defined by the following conventions:

• Heat ($q$): A positive value (+) means the system gains heat; a negative value (-) means the system releases heat.
• Work ($w$): A positive value (+) means the system gains energy from work; a negative value (-) means the system releases energy by doing work.

A common type of work in chemistry is PV work, which is work caused by a change in volume against an external pressure. For a system expanding against constant external pressure, the work is calculated as:

$w = -P \Delta V$

State Functions

Not all properties of a system are treated equally. State functions are properties that depend only on the current state of the system (its initial and final conditions) and not on the path taken to get there. Examples include internal energy ($E$), pressure ($P$), volume ($V$), and temperature ($T$).

Measuring Heat: Calorimetry

To measure the thermal energy exchanged during a reaction, scientists use calorimetry.

When measuring heat, we distinguish between two concepts:

1. Heat Capacity ($C$): The amount of heat required to raise the temperature of a given quantity of a substance by $1^{\circ}\mathrm{C}$.
2. Specific Heat Capacity ($C_s$): The amount of heat required to raise the temperature of one gram of a substance by $1^{\circ}\mathrm{C}$.

The formula for heat ($q$) is: $q = m C_s \Delta t$ Where $m$ is mass and $\Delta t$ is the change in temperature ($t_{\text{final}} - t_{\text{initial}}$).

Exam Tip: When two objects at different temperatures are placed in contact, heat flows from the hot object to the cold object until they reach thermal equilibrium. In these problems, remember that the heat lost by the hot object is equal to the heat gained by the cold object: $q_{\text{hot}} = -q_{\text{cold}}$.

Enthalpy ($\Delta H$)

While internal energy ($\Delta E$) is a useful concept, most chemical reactions occur at constant pressure (such as in an open beaker). For these processes, we use Enthalpy ($H$) to quantify heat flow. The change in enthalpy ($\Delta H$) is the heat involved in a reaction at constant pressure:

$\Delta H_{\text{reaction}} = q_{\text{reaction at constant pressure}}$

Exothermic vs. Endothermic Processes

Chemical reactions can either release or absorb energy, which we categorize into two types:

1. Exothermic Process: The system releases heat to the surroundings. The surroundings' temperature rises, and the enthalpy change is negative ($\Delta H < 0$).

   • Example: The combustion of methane.

     

2. Endothermic Process: The system absorbs heat from the surroundings. The surroundings' temperature drops, and the enthalpy change is positive ($\Delta H > 0$).

   • Example: The melting of ice.

     

Relationship between $\Delta H$ and $\Delta E$

The difference between enthalpy and internal energy is related to the work done by or on the system:

$\Delta H = \Delta E + P \Delta V$

For reactions involving gases, this difference can be significant. If a reaction produces a large volume of gas, $\Delta H$ and $\Delta E$ will differ more noticeably than in reactions occurring in aqueous solutions.

Calculating Enthalpy Changes

Hess's Law

Hess's Law states that the change in enthalpy for a stepwise process is the sum of the enthalpy changes of the individual steps. This allows us to calculate the $\Delta H$ of a complex reaction by breaking it down into simpler, known reactions.

Standard Enthalpy of Formation ($\Delta H_f^{\circ}$)

A very useful way to calculate the enthalpy of a reaction is by using the Standard Enthalpy of Formation. This is the enthalpy change for the reaction forming 1 mole of a pure compound from its constituent elements in their standard states (at 1 atm and usually $25^{\circ}\mathrm{C}$).

Important Note: The $\Delta H_f^{\circ}$ for any pure element in its standard state (such as $\mathrm{O}_2(g)$ or $\mathrm{C}(\text{graphite})$) is exactly $0 \text{ kJ/mol}$.

To find the standard enthalpy of a reaction ($\Delta H_{\text{rxn}}^{\circ}$), you subtract the sum of the enthalpies of the reactants from the sum of the enthalpies of the products:

$\Delta H_{\text{rxn}}^{\circ} = \sum n \Delta H_f^{\circ}(\text{products}) - \sum m \Delta H_f^{\circ}(\text{reactants})$

Worked Example: Combustion of Methane

To calculate the enthalpy change for the combustion of methane: $\mathrm{CH}_4(g) + 2\mathrm{O}_2(g) \rightarrow \mathrm{CO}_2(g) + 2\mathrm{H}_2\mathrm{O}(g)$

Using standard values from a table:

• $\Delta H_f^{\circ} [\mathrm{CO}_2(g)] = -393.5 \text{ kJ/mol}$
• $\Delta H_f^{\circ} [\mathrm{H}_2\mathrm{O}(g)] = -241.8 \text{ kJ/mol}$
• $\Delta H_f^{\circ} [\mathrm{CH}_4(g)] = -74.6 \text{ kJ/mol}$
• $\Delta H_f^{\circ} [\mathrm{O}_2(g)] = 0 \text{ kJ/mol}$

The calculation is:

\Delta H^{\circ} &= [(-393.5) + 2(-241.8)] - [(-74.6) + 2(0)] \\ \Delta H^{\circ} &= [-877.1] - [-74.6] \\ \Delta H^{\circ} &= -802.5 \text{ kJ} \end{aligned}$$ ![An enthalpy diagram illustrating the combustion of methane through decomposition and formation steps. This is useful for teaching thermochemistry and enthalpy changes in chemical reactions.]( &#36;&#36;&#36; &#36;&#36;&#36;