Chapter 6.3: Thermochemistry – Work, Internal Energy, and Calorimetry

Thermochemistry

Introduction

Thermochemistry is the study of the energy and heat associated with chemical reactions and physical transformations. It focuses on the transfer of energy between a system and its surroundings, particularly in the form of heat (q) and work (w).

Quantifying Work in Chemical Systems

Work and Energy Transfer

• Energy can be transferred between a system and its surroundings as heat (q) or work (w).

• The change in internal energy of a system is given by: $\Delta U = q + w$

• Work due to volume change at constant pressure is: $w = -P\Delta V$ where $\Delta V = V_{\text{final}} - V_{\text{initial}}$

• The negative sign indicates that when the system expands ($\Delta V > 0$), it does work on the surroundings, and energy leaves the system.

Example: Work Done in a Reaction

• For the reaction: $\mathrm{CH_4(g) + 2O_2(g) \rightarrow CO_2(g) + 2H_2O(l)}$

• At constant pressure and temperature (e.g., 298 K), the work done per mole of methane is calculated as: $w = -\Delta n_{\text{gas}}RT$ where $\Delta n_{\text{gas}} = n_{\text{final}} - n_{\text{initial}}$ (change in moles of gas), $R$ is the gas constant, and $T$ is temperature in Kelvin.

• For this reaction: $\Delta n_{\text{gas}} = (1 - 3) = -2$ $w = -(1 - 3)(8.314\,\text{J mol}^{-1}\text{K}^{-1})(298\,\text{K}) = 4955.14\,\text{J} = 4.96\,\text{kJ}$

Measuring $\Delta U$ for Chemical Reactions

Internal Energy Change at Constant Pressure and Volume

• At constant pressure: $\Delta U = q + w = q_p - P\Delta V$ where $q_p$ is the heat at constant pressure.

• At constant volume: $w = 0$ (since $\Delta V = 0$), so $\Delta U = q_v$ where $q_v$ is the heat at constant volume.

• Heat transfer can be measured using the formula: $q = mC_s\Delta T$ where $m$ is mass, $C_s$ is specific heat capacity, and $\Delta T$ is the temperature change.

Constant Volume Calorimetry

Bomb Calorimeter

• Constant volume calorimetry is typically performed using a bomb calorimeter, which is used to measure the heat of combustion of substances.

• The system (sample) is placed in a sealed container (the bomb) surrounded by water (the surroundings).

• The bomb is insulated to prevent heat exchange with the external environment.

• Since the volume is constant, no work is done ($w = 0$), and all energy change is measured as heat.

• The heat produced by the reaction increases the temperature of the water, which is measured to determine the energy change.

Calorimeter Calibration and Calculations

• Each calorimeter must be calibrated using a substance with a known heat of combustion.

• The heat capacity of the calorimeter ($C_{\text{cal}}$) is determined by: $C_{\text{cal}} = \frac{q_{\text{known}}}{\Delta T}$

• For a reaction, the heat absorbed by the calorimeter is: $q_{\text{cal}} = C_{\text{cal}} \times \Delta T$

• The heat of reaction at constant volume is: $q_v = -q_{\text{cal}}$

• The change in internal energy per mole is: $\Delta U = q_v$ $\Delta U_{\text{per mole}} = \frac{\Delta U}{n}$

Example: Bomb Calorimeter Calibration

• Given: 1.0215 g of benzoic acid (molar mass = 122.12 g/mol) burned, temperature rises by 2.538°C, $\Delta U_{\text{BA}} = -3222.0$ kJ/mol.

• Calculate $C_{\text{cal}}$: $q_{\text{BA}} = n \times \Delta U_{\text{BA}}$ $n = \frac{1.0215\,\text{g}}{122.12\,\text{g/mol}} = 0.00837\,\text{mol}$ $q_{\text{BA}} = 0.00837\,\text{mol} \times (-3222.0\,\text{kJ/mol}) = -26.97\,\text{kJ}$ $C_{\text{cal}} = \frac{-q_{\text{BA}}}{\Delta T} = \frac{26.97\,\text{kJ}}{2.538\,^{\circ}\text{C}} = 10.62\,\text{kJ}/^{\circ}\text{C}$

Enthalpy ($H$) and Enthalpy Change ($\Delta H$)

Definition and Properties

• Enthalpy ($H$) is a state function defined as: $H = U + PV$ where $U$ is internal energy, $P$ is pressure, and $V$ is volume.

• The change in enthalpy at constant pressure is: $\Delta H = \Delta U + P\Delta V$

• At constant pressure, the heat transferred is equal to the enthalpy change: $q_p = \Delta H$

• Enthalpy is a state function, meaning it depends only on the initial and final states, not the path taken.

Exothermic and Endothermic Processes

• If $\Delta H > 0$, the process is endothermic (absorbs heat).

• If $\Delta H < 0$, the process is exothermic (releases heat).

• The enthalpy change for a reaction is: $\Delta H = H_{\text{products}} - H_{\text{reactants}}$

Comparing $\Delta U$ and $\Delta H$

Relationship and Differences

• For most reactions not involving gases, $\Delta U \approx \Delta H$.

• For reactions involving gases, the difference is significant due to work done by gas expansion or compression.

• The relationship is: $\Delta H = \Delta U + \Delta n_{\text{gas}}RT$ where $\Delta n_{\text{gas}}$ is the change in moles of gas.

Summary Table: Key Thermochemical Quantities

Quantity	Symbol	Definition	Equation
Internal Energy Change	$\Delta U$	Change in total energy of a system	$\Delta U = q + w$
Work (at constant P)	$w$	Work done by system due to volume change	$w = -P\Delta V$
Enthalpy Change	$\Delta H$	Heat transferred at constant pressure	$\Delta H = \Delta U + P\Delta V$
Heat at Constant Volume	$q_v$	Heat measured in bomb calorimeter	$q_v = \Delta U$
Heat at Constant Pressure	$q_p$	Heat measured in coffee-cup calorimeter	$q_p = \Delta H$

Key Concepts and Practice

• Thermochemistry involves understanding how energy is transferred as heat and work during chemical reactions.

• Calorimetry is the experimental technique used to measure heat changes in chemical processes.

• State functions like $U$ and $H$ depend only on the state of the system, not the process used to reach that state.

• Practice problems often involve calculating $\Delta U$, $\Delta H$, and interpreting calorimetry data.