Thermodynamics

Thermodynamic Systems and the Concept of the Universe

• Types of Thermodynamic Systems

  • Open System: A system in which both energy and material can be exchanged with the surroundings. A typical example is a beaker on a benchtop that is open to the atmosphere and can be heated or cooled.

  • Closed System: A system that is able to exchange energy with its surroundings but cannot exchange material. A sealed pot, such as a pressure cooker, serves as an example.

  • Isolated System: A system that is completely sealed off. It cannot change its volume, cannot exchange material, and is insulated against any temperature change. A thermos flask is a representative example of an isolated system.

• The Concept of the Universe

  • In thermodynamics, the "universe" refers to the entire scope of the system under study and its surroundings.

  • In an isolated system like a thermos flask, the universe is limited to the contents of the flask because they are out of touch with anything else.

  • In an open system like a beaker on a bench, the universe includes the beaker and the entire atmosphere.

The First Law of Thermodynamics

• Foundational Principle

  • Energy cannot be created or destroyed; it only changes form or moves between parts of a system.

  • The law is effectively a conservation law stating that energy cannot disappear or appear without a flow occurring.

• Mathematical Representation

  • The change in the internal energy of a system ($\Delta U$) is achieved through two processes: heat flow ($q$) and work ($w$).

  • The formula is expressed as: $\Delta U = q - w$

• Sign Conventions for Energy Flow

  • Heat Flow ($q$): Defined as the flow from outside into the system.

    • When heat flows into the system, $q$ is a positive number.

    • When heat flows out of the system, $q$ is a negative number.

  • Work ($w$): Defined as the work done by the system on the surroundings.

    • When the system does positive work on the surroundings, the sign is negative in the energy equation because the energy of the system decreases.

• Historical Significance

  • The First Law was the last of the laws to be created but is considered the most profound.

  • It marked the point where the scientific community abandoned the "caloric" theory—the idea that heat was a mysterious, invisible fluid—and realized that heat flow is simply a form of energy exchange.

Mechanical Work and Volume Changes

• Pressure-Volume Work

  • A gas in a piston pushing against a constant pressure ($P$) performs work based on its change in volume ($\Delta V$).

  • The general definition of work in physics is force multipling displacement, which translates in thermodynamics to pressure multipling volume change: $w = P \times \triangle V$

  • Volume Change ($\Delta V$): The distance the piston travels multiplied by the area of the piston.

• Work and Energy Relationship

  • Expansion: During expansion, the volume increases ($\Delta V > 0$). Consequently, $w$ is positive, meaning work is done by the system on the surroundings. Energy leaves the system, leading to a decrease in internal energy ($\Delta U < 0$).

  • Compression: During compression, the volume decreases ($\Delta V < 0$). This results in negative work (work done on the system), and the internal energy of the system increases ($\Delta U > 0$).

• Other Forms of Work

  • Electrical Work: Charging, aligning, or polarizing materials.

  • Surface Work: Creating surfaces or breaking objects into smaller pieces.

  • Elastic Work: Stretching or straining materials.

Heat Capacity and Measurement

• The Problem of Measuring Energy Directlty

  • Energy is fundamental but nearly impossible to see or measure directly. Instead, we measure variables such as volume changes or temperature changes to calculate energy.

  • Even temperature is often measured indirectly; common thermometers measure the change in volume of a liquid to signify temperature change.

• Heat Capacity ($C$)

  • Heat flow into a system is proportional to the temperature change. The constant of proportionality is the heat capacity: $q = C \times \triangle T$

  • Heat capacity is the amount of heat required to raise the temperature of a system by one degree Celsius (or one Kelvin).

• Variables of Heat Capacity

  • Molar Heat Capacity ($C_m$): The heat capacity per mole of material. The units are Joules per Kelvin per mole ($J\,K^{-1}\,mol^{-1}$).

  • Specific Heat ($c$): The heat capacity per gram of material. The units are Joules per Kelvin per gram ($J\,K^{-1}\,g^{-1}$).

  • Calculating Heat Flow ($q$):

    • Using moles ($n$): $q = n \times C_m \times \triangle T$

    • Using mass ($m$): $q = m \times c \times \triangle T$

• Extensive vs. Intensive Properties

  • Extensive Quantities: Scaled by the size of the system. Examples include internal energy, volume, and heat capacity. Doubling the system doubles these quantities.

  • Intensive Quantities: Do not depend on the size of the system. Examples include temperature and pressure.

• Material Specifics

  • Metals generally have low heat capacities. For example, gold requires only $0.129\,J/g$ to raise its temperature by $1\,K$.

  • Liquid Water ($H_2O$) has one of the largest heat capacities, requiring significantly more energy (approximately 20 times more than gold or other metals) to achieve the same temperature rise.

  • Water's high heat capacity is due to its many "degrees of freedom"—vibrations, rotations, and rearrangements of hydrogen bonds—which allow it to store large amounts of energy.

Calorimetry and Phase Transitions

• Historical Calorimeters

  • The first calorimeters used ice. A reaction container was surrounded by ice; heat released by the reaction melted the ice into water, which was measured by volume in a bucket. Every volume of water corresponded to a specific amount of heat energy.

• Modern Calorimeters

  • Bomb Calorimeter: A sealed, reinforced container that measures heat changes at constant volume ($\Delta V = 0$). Total heat change is measured by a thermometer in a surrounding heat bath.

  • Bench-top Calorimeter: Open to the atmosphere, meaning measurements are taken at constant pressure ($\Delta P = 0$).

• Heat Capacity at Constant Pressure vs. Volume

  • Heat capacity at constant pressure ($C_p$) is usually larger than heat capacity at constant volume ($C_v$) because some of the heat energy added goes toward doing expansion work against the surroundings rather than just raising the temperature.

• Phase Transitions

  • Enthalpy of Fusion: The energy required to transition from solid to liquid. Melting is endothermic ($\Delta H > 0$).

  • Enthalpy of Vaporization: The energy required to transition from liquid to gas. Boiling is endothermic.

  • Sublimation: The direct transition from solid to gas. The enthalpy of sublimation is the sum of the enthalpy of fusion and the enthalpy of vaporization ($\Delta H_{sub} = \Delta H_{fus} + \Delta H_{vap}$).

  • Temperature Plateaus: During a phase transition (e.g., ice melting), adding heat does not raise the temperature. The temperature remains fixed at the melting or boiling point until the entire substance has changed phase.

Enthalpy ($H$)

• Definition and Utility

  • Enthalpy is a state function defined as: $H = U + PV$

  • For processes at constant pressure, the change in enthalpy is exactly equal to the heat flow: $\Delta H = q_p$

  • This is useful because heat flow is easily measured experimentally in open bench-top systems.

• Standard Enthalpies

  • Endothermic Process: $\Delta H > 0$. The system absorbs heat from the surroundings and feels cold. Example: dissolving ammonium chloride.

  • Exothermic Process: $\Delta H < 0$. The system releases heat to the surroundings and feels hot. Example: dissolving calcium chloride or combustion of butane.

  • Standard State: Denoted by a superscript zero ($^∘$). Refers to a substance's pure form at equilibrium at $298\,K$ ($25\,^‘C$) and $1\,atm$ pressure.

• Heats of Solution and Ionic Solids

  • Ionic solids have very high melting points because of strong Coulomb interactions in their crystal lattice. However, water can dissolve these lattices at room temperature.

  • The exothermic heat of solution for some salts indicates that ions are actually more stable floating in water than they were in a perfectly packed crystal lattice. This is a "magic trick" unique to water.

Predictors of Reaction Enthalpy

• Hess's Law

  • The total enthalpy change for a process is the sum of the enthalpy changes for the individual individual steps, regardless of the path taken. This relies on enthalpy being a state function.

• Standard Enthalpy of Formation ($\Delta H_f^∘$)

  • The enthalpy change when one mole of a molecule is formed from its component elements in their standard states.

  • The standard heat of formation of an element in its correct standard state (e.g., $H_2(g)$, $O_2(g)$, $C(graphite)$, $Hg(l)$, $I_2(s)$) is always zero.

  • Calculating Enthalpy of Reaction: $\Delta H_{rxn}^∘ = \sum \Delta H_{f,products}^∘ - \sum \Delta H_{f,reactants}^∘$

• Bond Enthalpies

  • Bond Breaking: Always an endothermic process; energy must be put in to break a bond.

  • Bond Making: Always an exothermic process; energy is released when bonds form.

  • While tabulated average bond enthalpies can estimate $\Delta H_{rxn}$, they are not as accurate as using heats of formation because the actual energy of a bond (e.g., $C-H$) varies slightly depending on the specific molecule it is in.

Fuel Types and Energy Resources

• Hydrocarbon Combustion

  • Combustion involves oxidizing organic material to release heat, which drives machinery or biological systems.

  • The energy released by burning fuels comes primarily from the production of Carbon Dioxide ($CO_2$) and Water ($H_2O$), which are very stable, low-energy molecules.

• Hydrogen as Fuel

  • Benefits: Releases energy without producing $CO_2$; the byproduct is water vapor, which is greenhouse-neutral.

  • Drawbacks:

    • Generation: Producing hydrogen gas (e.g., electrolysis) requires significant energy.

    • Storage: Hydrogen has a very low energy density by volume. A volume of hydrogen equivalent to a liter of petrol would fill half a lecture hall at standard pressure.

    • Technical Obstacles: Compression requires high-pressure, potentially explosive cylinders. Metals can absorb hydrogen like a sponge, but getting them to release it efficiently is a currently unsolved challenge.

Entropy ($S$) and the Second Law

• The Limit of the First Law

  • The First Law allows for heat to flow from a cold object to a hot object, as long as energy is conserved. However, we know empirically that heat always flows from hot to cold spontaneously.

• Definition of Entropy

  • Entropy is defined by the heat flow at a given temperature: $\triangle S = \frac{q}{T}$

  • Microscopically, Ludwig Boltzmann defined entropy ($S = k \ln(W)$) as the number of ways ($W$) energy can be distributed among the atoms and molecules of a system.

• The Second Law of Thermodynamics

  • For any spontaneous process, the total entropy of the universe ($S_{univ} = S_{sys} + S_{surr}$) must increase ($\Delta S_{total} > 0$).

  • At equilibrium, $\Delta S_{total} = 0$.

• The Third Law of Thermodynamics

  • A perfect crystal at absolute zero temperature ($0\,K$) has zero entropy. This allows for the calculation of absolute entropy values as temperature increases through solid, liquid, and gas phases.

Gibbs Free Energy ($G$)

• Mathematical Origin

  • The Scotsmen and other practical physicists of the 19th century preferred energy-like quantities over abstract entropy. By multiplying the Second Law expression by $-T$, the Gibbs Free Energy was derived.

  • Formula: $G = H - TS$

  • Gibbs Free Energy is expressed entirely in terms of the system's properties, removing the need to measure the surroundings.

• Criteria for Spontaneity

  • A process is spontaneous if the change in Gibbs Free Energy is negative ($\Delta G < 0$).

  • Temperature Dependence:

    • Exothermic ($\Delta H < 0$) with Increasing Entropy ($\Delta S > 0$): Always spontaneous at all temperatures.

    • Endothermic ($\Delta H > 0$) with Decreasing Entropy ($\Delta S < 0$): Never spontaneous at any temperature.

    • Exothermic ($\Delta H < 0$) with Decreasing Entropy ($\Delta S < 0$): Spontaneous only at low temperatures.

    • Endothermic ($\Delta H > 0$) with Increasing Entropy ($\Delta S > 0$): Spontaneous only at high temperatures.

• Reaction Coupling

  • Processes with a positive $\Delta G$ (non-spontaneous) can be driven forward by coupling them to another reaction with a much larger negative $\Delta G$.

  • Industrial Smelting: Extracting copper from copper oxide has a positive $\Delta G$. By coupling it to the combustion of carbon (charcoal), the net process becomes negative and spontaneous.

  • Biology: Many metabolic processes are powered by coupling them to the dephosphorylation of ATP (Adenosine Triphosphate).

Questions & Discussion

• Question: How does a thermometer work? What do you actually measure?

• Response: You measure a change in volume. Even temperature is measured indirectly through other variables.

• Question: Are there instances where you can access energy directly?

• Response: Yes, as with black body radiation. If you use photomultipliers, you can measure the energies of incident photons directly, but this is less common in everyday measurements.

• Question: Why do different materials like water and gold have different heat capacities?

• Response: It depends on how many degrees of freedom the material has to store energy. In simple metals, energy is stored only in vibrations. Water can store energy in vibrations, rotations, translational movement, and the configuration of hydrogen bonds.

• Question: Why are $C_p$ and $C_v$ different?

• Response: If you allow the system to expand (constant pressure), some of the heat put in is used to do work on the surroundings rather than solely raising the temperature. Therefore, you need more heat to achieve the same temperature rise.

• Question: Does breaking a bond release energy?

• Response: Never. You must always put energy in to break a bond. Only bond making releases energy. An overall reaction is only exothermic if the energy released by forming new bonds is greater than the energy required to break the old ones.

• Question: When does heat stop flowing between two objects?

• Response: When they reach the same temperature. The final equilibrium temperature ($T_{eq}$) can be calculated from the initial temperatures and heat capacities: $T_{eq} = \frac{C_{hot}T_{hot} + C_{cold}T_{cold}}{C_{hot} + C_{cold}}$.

• Question: What is an unanswered question like?

• Response: An unanswered question goes off like a banana at the bottom of a school bag; it leads to moldy old problems that nobody got around to asking. It is better to get them out of your system.