Adiabatic Processes: Thermodynamics

Adiabatic Processes

• An adiabatic process is one where there is no heat flow into or out of the system, q=0
• The temperature of a blob of dry air that expands adiabatically changes by about 10°C for each kilometer of elevation.
• wad = CvdT = -pex dV
• Cv dT = −nRTV−1
• Cv dT/T = −nR dV/V
• Cv ∫(Ti to Tf) dT/T = −nR ∫(Vi to Vf) dV/V
• Cv ln(Tf/Ti) = −nR ln(Vf/Vi)
• CV,m/R ln(Tf/Ti) = ln(Vi/Vf)
• (Tf/Ti)^(Cv,m/R) = (Vi/Vf)
• ViTi^c = VfTf^c; Relates T and V for adiabatic process, where c = Cv,m/R
• The work done and ∆U can be determined when 0.02 mol ammonia initially at 298 K is used in a reversible adiabatic expansion from 0.50 to 2.0 L
• Cvm(NH3) = 27.2 J/Kmol
• Reversible adiabatic expansion: w = nCv,m∆T
• Calculations require values of Ti, Tf, and Cv,m
• Tf = Ti X (Vi/Vf)^(1/c)
• Cv,m ln(Tf/Ti) = -R ln(Vf/Vi)
• Using Cp,m - Cv,m = R (relationship for ideal gas)
• Cp,m/Cv,m - 1 = R/Cv,m if: γ = Cp,m/Cv,m
• Thus γ − 1 = R/Cv,m Giving pγVi = pfVγ
• Relates p and V for adiabatic process.
• Two isotherms can be connected by an adiabat
• w1→2 = −∫(V1 to V2) P(V,T)dV
• w1→2 = −∫(V1 to V2) P1V1^γ / V^γ dV = −P1Vγ(V^(−γ+1) / (−γ+1)|(V1 to V2)
• w1→2 = P1V1γ /(γ − 1) (1 / (V2^(γ−1) − 1/V1^(γ−1))
• pγVi = pfVγ; γ = Cp,m/Cv,m
• The final pressure can be calculated when a sample of Ar which γ = 5/3 expands reversibly and adiabatically to twice its initial volume.
• pf = pi × (Vi/Vf)^γ

Thermochemistry

• Thermochemistry: the study of energy transferred as heat during reactions.
• Chemical reactions result in the exchange of energy between the system and the surroundings.
• Calorimetry can measure the energy supplied or discarded as heat by a reaction and can identify q with internal energy or enthalpy.
• Exothermic processes release energy by heating the surroundings.
• Endothermic processes absorb energy by cooling the surroundings.
• Changes in enthalpy are normally reported for processes taking place under a set of standard conditions.
• The standard enthalpy change, ΔH°, is for a process in which the initial and final substances are in their standard states.
• The standard state of a substance is its pure form at 1 bar at a specified temperature.
• Standard enthalpy change for a reaction or a physical process is the difference between reactants and products in their standard states.
• The standard enthalpy of vaporization, ∆vapH°, is the enthalpy change per mole when a pure liquid at 1 bar vaporizes to a gas at 1 bar.
• H2O(l) → H2O(g): ∆vapH°(373K) = +40.66kJmol-1
• Standard enthalpy of transition: The standard enthalpy change that accompanies a change of physical state and is denoted ∆trsH°.
  • H2O(s) → H2O(l): ∆fusH°
  • H2O(l) → H2O(g): ∆vapH°
  • H2O(s) → H2O(g): ∆subH°
• Since H is a state function, ∆subH° can be obtained via fusion then vaporization.
  • H2O(s) → H2O(l): ∆fusH°
  • H2O(l) → H2O(g): ∆vapH°
• Overall: H2O(s) → H2O(g): and ∆subH° = ∆fusH° + ∆vapH°
• Standard enthalpy changes of a forward process relates to its reverse.
• ΔH°(A → B) = −ΔH°(B → A)
• The enthalpy change is positive for vaporization of water: H2O(l) → H2O(g): ∆vapH° = +44kJmol−1
• The enthalpy change is negative for condensation of water: H2O(g) → H2O(l): ∆conH° = −44kJmol−1
• There are two ways of reporting the change in enthalpy that accompanies a chemical reaction.
  • Writing the thermochemical equation: the chemical equation and the corresponding change in standard enthalpy
  • CH4(g) + 2O2(g) → CO2(g) + 2H2O(l) ΔH° =-890kJ
  • ∆He is the change in enthalpy when reactants in their standard states change to products in their standard states.
• The standard value (ΔH°) refers to the reaction in which 1 mol CH4 in the form of pure methane gas at 1 bar reacts completely with 2 mol O2 in the form of pure oxygen gas to produce 1 mol CO2 as pure carbon dioxide at 1 bar and 2 mol H2O as pure liquid water at 1bar, at 298 K.
• Pure, separate reactants in their standard states → pure, separate products in their standard states
• ii. Writing the chemical equation and then report the standard reaction enthalpy, ∆rH°. Thus, for the combustion reaction CH4(g) + 2O2(g) → CO2(g) + 2H2O(l) ΔrH° = −890 kJ mol-1
• Consider a general reaction: 2A + B → 3C + D The standard reaction enthalpy is given by:
• ∆rH° = {3Hm(C) + Hm(D) → Hm(B) + 2Hm(A)}
• Hom(J) is the standard molar enthalpy of species J at the temperature of interest.
• ∆rH° = ΣνHm(products) − ΣνHm(reactants)
• The standard enthalpy of formation, (∆fH°), of a substance is the standard reaction enthalpy for the formation of the compound from its elements in their reference states
• The reference state of an element is its most stable state at the specified temperature and 1 bar.
• Standard enthalpies of formation, (∆fH°), are measured under standard conditions (25°C and 1.00 bar pressure).
• If there is more than one state for a substance under standard conditions, the more stable one is used.
• Standard enthalpy of formation of the most stable form of an element is zero.