Electrostatic Potential Energy, Thermodynamics, and Enthalpy

Electrostatic Forces and Coulomb's Law

• Fundamental Forces at the Atomic Scale:

  • Gravity is familiar in everyday macroscopic life, but it is entirely negligible on the molecular and atomic levels because molecules have extremely small masses and gravity is an exceptionally weak force at atomic scales.

  • The force that predominantly governs atomic and molecular interactions is the electromagnetic force.

• Coulomb's Law Equation:

  • Electrostatic potential energy ($E$) between two charged species is quantified by Coulomb's law:     $E = k \frac{q_1 q_2}{r}$

  • Where:

  • $E$ is the electrostatic potential energy.

  • $k$ is a proportionality constant.

  • $q_1$ and $q_2$ are the charges of the two interacting species (e.g., for two protons, $q_1 = +1$ and $q_2 = +1$; for an electron and a proton, $q_1 = -1$ and $q_2 = +1$).

  • $r$ is the distance between the centers of the two charged particles.

• Distinction Between Force and Energy Equations:

  • In physics, Coulomb's Law for force features an $r^2$ term in the denominator:     $F = k \frac{q_1 q_2}{r^2}$

  • In chemistry and thermodynamics, the equation for electrostatic potential energy features $r$ raised to the first power ($r^1$) because potential energy is derived by integrating the force function with respect to distance ($r$).

• Graphical Behavior of Electrostatic Potential Energy:

  • Particles with Like Charges (e.g., two protons or two electrons):

  • The product $q_1 q_2$ is always positive ($(+1) \times (+1) = +1$ or $(-1) \times (-1) = +1$).

  • Electrostatic potential energy ($E$) is positive at all distances $r$.

  • The plot of $E$ versus $r$ mirrors the inverse function $y = \frac{1}{x}$.

  • At distances near zero ($r \rightarrow 0$), the potential energy is extremely high. As distance $r$ increases, energy decreases asymptotically toward zero ($E \rightarrow 0$).

  • Physical systems naturally tend toward the state of lowest possible energy. Therefore, like-charged particles experience electrostatic repulsion to move as far apart as possible, lowering their potential energy.

  • Particles with Opposite Charges (e.g., a proton and an electron):

  • The product $q_1 q_2$ is always negative ($(+1) \times (-1) = -1$).

  • Electrostatic potential energy ($E$) is negative at all distances $r$.

  • The graph of $E$ versus $r$ is a vertical reflection (mirror image across the distance axis) of the like-charge curve.

  • The lowest energy state occurs when the particles are closest together ($r \rightarrow 0$ yields the most negative energy value).

  • Consequently, oppositely charged particles attract one another, moving closer together to lower their potential energy.

Systems, Surroundings, and Internal Energy

• Thermodynamic Definitions:

  • System: Any specific, defined subset of matter or material chosen for examination or study.

  • Surroundings: Everything else in the universe outside the defined system.

• Internal Energy ($E$):

  • Internal energy ($E$) is defined as the sum total of all kinetic energy ($KE$) and potential energy ($PE$) contained within a system:     $E = KE + PE$

  • The total energy of the universe is conserved and remains constant.

  • The internal energy of a system changes as energy is transferred across the boundary between the system and its surroundings.

• Change in Internal Energy ($\Delta E$):

  • The symbol $\Delta$ (delta) denotes a change in a thermodynamic quantity, defined as the final state value minus the initial state value:     $\Delta E = E_{\text{final}} - E_{\text{initial}}$

  • Negative Change (\Delta E < 0): The system lost energy to the surroundings. The initial state had higher energy than the final state.

  • Positive Change (\Delta E > 0): The system gained/absorbed energy from the surroundings. The initial state had lower energy than the final state.

State Functions vs. Path Functions

• Energy Transfer Pathways: Heat ($q$) and Work ($w$):

  • Energy is exchanged between a system and its surroundings through two primary pathways: heat ($q$) and work ($w$).

  • Heat ($q$): Energy transferred due to thermal interactions.

  • q < 0 (negative): System loses heat to surroundings.

  • q > 0 (positive): System absorbs heat from surroundings.

  • Work ($w$): Energy transfer resulting from a force exerted over a distance ($W = F \times d$). For example, a chemical reaction expanding in size pushes back against external atmospheric pressure, performing work.

  • w < 0 (negative): System performs work on the surroundings (loses energy).

  • w > 0 (positive): Surroundings perform work on the system (system gains energy).

  • Total Energy Change Equation:     $\Delta E = q + w$

• State Functions:

  • A state function is a parameter describing the current condition or state of a system, independent of the path or mechanism taken to reach that condition.

  • State functions are abbreviated with capital letters (e.g., Internal Energy $E$, Temperature $T$, Pressure $P$, Enthalpy $H$).

  • Beaker Example: A beaker of hot water cooling down (q < 0) and a beaker of cold water warming up (q > 0) brought to identical room temperatures possess identical internal energy. At room temperature, they are completely indistinguishable from one another, providing no information about how they reached that state.

  • Mountain Elevation Analogy: Elevation above sea level (e.g., Half Dome / Afton at Yosemite) is a state function. The elevation change ($\Delta \text{elevation} = \text{elevation}<em>{\text{final}} - \text{elevation}</em>{\text{initial}}$) between base and summit is identical regardless of taking a steep right-side trail or a winding forested path.

• Path Functions:

  • A path function is a quantity whose value depends on the specific path taken between states.

  • Path functions are abbreviated with lowercase letters (e.g., heat $q$, work $w$).

  • Systems do not possess a fixed quantity of heat or work; rather, $q$ and $w$ describe the process of energy flow during a state transition.

Enthalpy and Thermochemistry

• Definition of Enthalpy ($H$):

  • Enthalpy ($H$) represents the capacity of a system to release energy in the form of heat.

  • Enthalpy change ($\Delta H$) is defined as the heat transferred under constant pressure ($q_p$):     $\Delta H = q_p$

  • Most ambient chemical reactions take place in open containers under constant atmospheric pressure ($1\,\text{atm}$).

• Relationship Between Enthalpy ($\Delta H$) and Internal Energy ($\Delta E$):

  • The mathematical relationship connecting internal energy and enthalpy is:     $\Delta E = \Delta H - P\Delta V$

  • Where $P$ is atmospheric pressure and $\Delta V$ is the change in system volume ($\Delta V = V_{\text{final}} - V_{\text{initial}}$).

  • $\Delta H$ is a state function (denoted by capital $H$).

  • For most chemical reactions, volume changes are negligible ($\Delta V \approx 0$). Consequently, $-P\Delta V \approx 0$, making $\Delta E \approx \Delta H$. Therefore, terms like enthalpy and energy are often used interchangeably in general chemistry.

• Exothermic vs. Endothermic Processes:

  • Exothermic Process:

  • \Delta H < 0 ($q_p$ is negative).

  • Heat exits the system into the surroundings.

  • Surroundings get hotter.

  • Potential vs. Kinetic Energy: In exothermic combustion (e.g., burning wood in a fireplace), potential energy stored in chemical bonds is converted into kinetic energy and lost as heat. The system does not get cold; heat releases rapidly, raising temperature.

  • Endothermic Process:

  • \Delta H > 0 ($q_p$ is positive).

  • Heat enters the system from the surroundings.

  • Surroundings get colder.

  • Thermal kinetic energy from surroundings is absorbed and converted into system potential energy.

Demonstration: Rocket Fuel Combustion

• Reaction Overview:

  • A rocket fuel combustion demonstration burning table sugar (sucrose, $\text{C}_{12}\text{H}_{22}\text{O}_{11}$) using a solid oxidant, potassium chlorate ($\text{KClO}_3$), rather than gaseous oxygen.

  • Reaction components:     $\text{C}_{12}\text{H}_{22}\text{O}_{11}(s) + \text{KClO}_3(s) \rightarrow \text{CO}_2(g) + \text{H}_2\text{O}(g) + \text{KCl}(s)$

• Experimental Execution:

  • Premixed solid sucrose and solid $\text{KClO}_3$ were placed inside a glass vial embedded in a sand bath.

  • Initiation: A drop or two of concentrated sulfuric acid ($\text{H}_2\text{SO}_4$) was added to supply activation heat and initiate combustion.

• Thermodynamic Analysis:

  • The reaction started with a large amount of chemical potential energy stored in reactant bonds and intermolecular interactions.

  • Potential energy was converted to products and released as heat (q < 0).

  • Because the process occurred under constant pressure, \Delta H = q_p < 0.

  • The reaction is exothermic, rapidly transferring thermal energy into the surroundings.

Questions, Dialogue, and Audience Interaction

• Class Logistics and Advice:

  • Question: What time does class end?

  • Response: Class ends at 04:20 PM or 04:25 PM (time during query was 03:57 PM).

  • Note-taking Strategy: Blank lecture slides are posted prior to class, and completed slides are posted after class. Students are advised to download and annotate the blank slides ahead of time.

• Potassium Chloride ($\text{KCl}$) Safety Discussion:

  • Question: Is $ KudCl$ poisonous or dangerous in the air?

  • Response: Inhaling tiny amounts in the air from burning sugar is harmless. Potassium ($\text{K}^+$) is an essential nutrient required for physiological cellular voltage potentials. However, injecting large quantities of $\text{KCl}$ directly into the bloodstream disrupts the cellular membrane potential across muscle tissue (including cardiac muscle) and is lethal (used in lethal injection procedures).

• Class Personnel & Assistance:

  • Faculty / Instructor: Dr. Gardner.

  • Students / Learning Assistants identified: Lach (spelled L-A-K-S-A) and Olivia (Liv).

  • TA Office Hours: Monday through Friday, 9:00 AM to 5:30 PM in the basement of Whitmore Lab.

Problem Solving and Conceptual Applications

• Top Hat Question 1: Ranking Electrostatic Potential Energy:

  • Problem: Rank three pairs of charged particles situated at distance $d$ in order of increasing electrostatic potential energy (lowest/most negative to highest/most positive).

  • Pair 1: $q_1 = -1$, $q_2 = +1$

  • Pair 2: $q_1 = -3$, $q_2 = -1$

  • Pair 3: $q_1 = -2$, $q_2 = +1$

  • Derivation & Calculation:

  1. Apply Coulomb's law: $E = k \frac{q_1 q_2}{d}$.

  2. For Pair 1: $E_1 = k \frac{(-1)(+1)}{d} = -\frac{k}{d}$.

  3. For Pair 2: $E_2 = k \frac{(-3)(-1)}{d} = \frac{3k}{d}$.

  4. For Pair 3: $E_3 = k \frac{(-2)(+1)}{d} = -\frac{2k}{d}$.

  5. Rank on a number line from lowest (most negative) to highest (most positive):        -\frac{2k}{d} < -\frac{k}{d} < \frac{3k}{d}

  • Final Ranking: Pair 3 < Pair 1 < Pair 2 (Answer choice E).

• Top Hat Question 2: System Thermodynamics Analysis:

  • Problem: A system performs no work ($w = 0$) but loses heat (q < 0). Potential energy ($PE$) remains constant. Evaluate the following statements:

  1. Statement 1: \Delta E < 0

  2. Statement 2: q < 0

  3. Statement 3: Kinetic energy ($KE$) decreased

  • Step-by-Step Proof:

  1. Evaluate Statement 2: Heat is lost by the system, so q < 0 is true.

  2. Evaluate Statement 1: Substitute $w = 0$ into $\Delta E = q + w$:        $\Delta E = q + 0 = q$        Since q < 0, \Delta E < 0 must be true.

  3. Evaluate Statement 3: System internal energy is $\Delta E = \Delta KE + \Delta PE$. Since $\Delta PE = 0$:        $\Delta E = \Delta KE$        Because \Delta E < 0, $\Delta KE$ must also be negative (\Delta KE < 0), meaning kinetic energy decreased.

  • Conclusion: Statements 1, 2, and 3 are all correct.