Review#
1. Checklist of Key Concepts#
Section 3.1: Conservation of Energy#
Overview
Energy conservation underlies all of thermodynamics.
Early demonstrations (e.g., Bernoulli, Euler) merged Newton’s mechanics with the idea that total mechanical energy remains constant in an isolated system.
Mechanical Equivalent of Heat
Joule’s experiments showed that mechanical work can be converted into heat in a fixed, quantitative ratio.
For water: \(\text{MEH} \approx 4.18\,\mathrm{J}\,\mathrm{g}^{-1}\,^\circ\mathrm{C}^{-1}\), establishing heat as another form of energy transfer.
First Law of Thermodynamics
Statement: \(\Delta U = q + w\), where \(q\) is heat absorbed by the system, \(w\) is work done on the system.
Path vs. State Functions: \(U\) is a state function (depends only on the current state), while \(q\) and \(w\) are path functions (depend on how the process is carried out).
Sign Convention: in chemistry, \(w>0\) if work is done on the system.
Types of Work
Generalized form: \(\delta w = \text{(generalized force)} \times \text{(generalized displacement)}\).
Common examples in thermodynamics:
\(P\,dV\) work: compression/expansion of a gas (\(\delta w = -P\,dV\)).
Surface tension: \(\delta w = -\gamma\,dA\).
Hooke’s law: \(\delta w = k\,(x - x_0)\,dx\).
Example calculations:
Expansion against constant pressure.
Inflating a bubble with surface tension.
Stretching a spring-like fiber obeying Hooke’s law.
Section 3.2: Applications of the First Law#
Thermodynamic Processes
Quasi-static: carried out slowly enough for the system to remain in (near) equilibrium at each step.
Reversible: idealized, quasi-static + no dissipative losses; can be undone with no net change in system or surroundings.
Irreversible: any real process that breaks one or more reversibility conditions.
How to Apply the First Law
Stepwise Procedure:
Choose two independent variables (e.g., \(V, T\) or \(P, T\)).
Rewrite \(\delta q\) and \(\delta w\) in terms of differentials of those variables.
Apply relevant constraints (e.g., isobaric, isochoric, isothermal, adiabatic).
Use the system’s equation of state (e.g., \(PV=nRT\) for an ideal gas).
Integrate or sum changes to find total \(q\), \(w\), and \(\Delta U\).
Using \(V\) and \(T\) as Independent Variables
First Law: \(\delta q = dU + P\,dV\).
For an ideal gas (\(\partial U/\partial V)_T = 0\)):
\[ \delta q = \left(\frac{\partial U}{\partial T}\right)_V dT + P\,dV.\]Special processes:
Isochoric (\(dV=0\)): \(q = \int C_V\,dT\).
Isothermal (\(dT=0\)): \(q = \int P\,dV = nRT \ln(V_2/V_1)\).
Adiabatic (\(q=0\)): \(\Delta U = w\). For a monoatomic ideal gas, \(T\,V^{\tfrac{2}{3}}=\text{const}\).
Microscopic Interpretation
Internal Energy: \(U = \sum_i p_i E_i\).
Heat (\(q\)) changes probabilities \(p_i\).
Work (\(w\)) shifts the energy levels \(E_i\) themselves (e.g., compressing the container changes spacing of quantum states).
This perspective unifies macroscopic thermodynamics with microscopic statistical mechanics.
Section 3.3: Enthalpy#
Definition of Enthalpy
\(H = U + PV\).
For a process at constant pressure, \(\delta q_P = dH\).
\(\Delta H\) and Heat at Constant Pressure
\(\Delta H = q_P\). Calorimetry at \(P=\text{const}\) measures enthalpy changes directly.
Heat Capacity at Constant Pressure: \(C_P = \bigl(\tfrac{\partial H}{\partial T}\bigr)_P\).
Standard Enthalpies
Standard State: \(P^\circ=1\text{ bar}\); tabulated data often at \(298.15\,\mathrm{K}\).
Standard Enthalpy of Formation, \(\Delta H_f^\circ\): enthalpy change to form 1 mole of a compound from its elements in their standard states.
By convention, \(\Delta H_f^\circ=0\) for any element in its standard state.
Standard Enthalpy of Reaction, \(\Delta H_{\mathrm{rxn}}^\circ\):
\[ \Delta H_{\mathrm{rxn}}^\circ \;=\;\sum_{\text{products}} \nu_p H_p^\circ \;-\;\sum_{\text{reactants}} \nu_r H_r^\circ.\]Hess’s Law: enthalpy changes are path independent; \(\Delta H\) of an overall reaction is the algebraic sum of enthalpy changes for its steps.
2. Checklist of Most Important Equations#
First Law of Thermodynamics
\[\Delta U \;=\; q \;+\; w,\]\(q\): heat absorbed by the system.
\(w\): work done on the system.
Differential form: \(dU = \delta q + \delta w.\)
Pressure–Volume Work (Constant External \(P\))
\[w \;=\; -\,P_{\text{ext}}\;\bigl(V_f - V_i\bigr).\]Sign convention: expansion (\(\Delta V>0\)) \(\Rightarrow\) \(w<0\) (system does work on surroundings).
Heat for Isochoric Process
\[q_{V} \;=\; \int C_V \, dT \;\;\Longrightarrow\;\; q_{V} = C_V\,\Delta T \;\;\text{if }C_V\approx\text{const.}\]Heat for Isothermal Process (Ideal Gas)
\[q_{\text{isoT}} \;=\; \int_{V_i}^{V_f} P\,dV \;=\; n\,R\,T\,\ln{\bigl(\tfrac{V_f}{V_i}\bigr)}.\]Adiabatic Condition (No Heat Exchange)
\[q_{\text{adi}} = 0 \;\;\Longrightarrow\;\; \Delta U = w.\]For a monoatomic ideal gas: \(T\,V^{\,2/3} = \text{constant}.\)
Enthalpy Definition
\[H \;=\; U + P\,V.\]At constant \(P\): \(\Delta H = q_P.\)
Standard Enthalpy of Reaction
\[\Delta H_{\mathrm{rxn}}^\circ \;=\; \sum_{p} \nu_p\,H_p^\circ \;-\; \sum_{r} \nu_r\,H_r^\circ.\]Hess’s Law and enthalpies of formation provide a straightforward way to compute \(\Delta H_{\mathrm{rxn}}^\circ\).