Introduction to Thermodynamics

• Thermodynamics is concerned with the changes in internal energy during transformations.
• These changes can be measured through heat and/or work exchanges.

Total Energy of a System

• Defined as the sum of macroscopic and microscopic kinetic and potential energies: Total E= (Ec+ Ep)macroscopic +(Ec+Ep)microscopic.
• Macroscopic kinetic energy (Ec) corresponds to the overall movement.
• Macroscopic potential energy (Ep) corresponds to interactions between the system and its external environment (e.g., gravity).
• Microscopic kinetic energy (Ec) arises from thermal agitation.
• Microscopic potential energy (Ep) involves interactions between elements within the system, such as infra-molecular and inter-molecular forces (e.g., hydrogen bonding).
• Various energy forms at the macroscopic/microscopic level can convert into each other, and can be exchanged with other systems or the external environment.

Internal Energy (U)

• Characterizes the energy content of matter and equals the sum of all energies possessed by the constituents: nucleus, electrons, atoms, and molecules.
• Includes kinetic energies from particle movements (translation, vibration, rotation).
• Includes potential energies from interactions between particles (attractions/repulsions between electrons and the nucleus, ions, dipole-dipole interactions, etc.).
• Internal energy exists even when a body is at rest from a macroscopic viewpoint due to persistent agitation (Ec) and interaction (Ep) of its microscopic particles.
• It is impossible to determine the exact sum of microscopic energies.

Energy Variation

• In a transformation between states, the total energy variation is defined as ΔE Total = Δ Ecg + Δ Epg + Δ Eci + Δ Epi
• ∆ Eci + ∆ Epi represents internal energy variation.
• For a stationary system with no macroscopic motion, ΔE = Δ U.
• The total energy exchanged as heat (Q) and mechanical work (W) equals the algebraic sum (Q + W) and is associated with the variation of a state function.
• Variation of internal energy is a state function, while work and heat are not.

Variation of Internal Energy (ΔU)

• For a closed system transforming from an initial state (Ei) to a final state (Ef) along different paths: (Q1+W1) = (Q2+W2) = (Q3+W3) = Cte = | Q + W |EfEi W1≠W2≠........≠W i and Q1≠Q2≠........≠Q.
• The notation |Q + W| corresponds to the total quantity of thermal energy and work exchanged by the system between the initial state Ei and the final state Ef.
• It is not an absolute value

Reformulating The First Principle

• There is a state function called internal energy, U.
• The variation ∆U of this energy during a transformation is equal to the sum of the work and the heat exchanged with the external environment.
• For an infinitesimal variation: dU = δQ + δW

Important Note

• dU is an exact total differential with mathematical meaning.
• δQ and δW are not exact total differentials, representing elementary exchanges of work and heat when U varies from the elementary quantity dU.

First Law of Thermodynamics Statement

• Also known as the principle of conservation of energy.
• All formulations are equivalent and linked to the conservative nature of energy.
• The energy of the system is conserved during transformations.
• The energy of the system is only transformed from one form to another.
• The energy of an isolated system remains constant (∆U= 0).
• Non-isolated system energy can vary through energy exchange (Q, W) with the external environment during a transformation.
• Internal energy variation during a transformation equals the algebraic sum of energies exchanged (W + Q).
• The internal energy of the system varies during the transformation between initial and final states.
• The total energy exchanged during a transformation, in heat (Q) and mechanical work (W), equals their algebraic sum (Q + W).

Internal Energy of an Ideal Gas

• Defined by fixing two of the state variables P, T and V.
• Variables adapted to internal energy U are T and V.
• U is a state function, its differential is written: dU = (∂U/∂V)T dV + (∂U/∂T)v dT
• James Joule showed in 1834 that internal energy of an ideal gas is independent of its volume at any temperature.

Joule's Experiment

• A container made of two parts (A and B) connected by a tap, immersed in a calorimeter with water, surrounded by an adiabatic enclosure.
• Initially, A contains the gas, and B is empty.
• On opening the tap, the gas spreads, pressures balance, and ΔT = 0.
• T = cte → Q = 0; W = 0 (Pext = 0).
• Hence: ΔU = U2 - U1 = Q + W = 0 ⇒ U gas = Cte ≠ f(V) and U gas = Cte ≠ f(P).
• The internal energy of an ideal gas depends only on temperature.
• T = Cte ⇒ ∆U gaz = 0 ⇒ U gaz = f(T); dU = (∂U/∂T)v dT= CvdT =ncdT
• For undergoing transformation by the ideal gas: ΔU = ∫ CvdT = ∫ n cydT = n cy ΔΤ
• Conclusion: ΔU = CAT =ncy∆T (Joule's first law).
• Thermal capacity at constant volume: total energy when macroscopically at rest without external action.
• For sufficiently diluted gas, Cv values are close to:
  • 3nR/2 (monatomic gases like He, Ne, Ar).
  • 5nR/2 (diatomic gases like O2, Cl2, N2).

Cyclic Transformation (A→A)

• The system returns to its initial state U2 Etat initial A Etat final B U1 ⇒ (Q1 + W₁) = -(Q2+ W2)

Isothermal Transformation (ΔT=0)

• According to Joule's first law: whatever the transformation undergone by the ideal gas: ∆U = Q + W = C v∆T = n c v ∆T = 0 since_∆T = 0(T = cte)
• ⇒ Q = -W
• For an isothermal evolution reversible between two states 1 and 2: P1V 1 = P2V 2
• Q =− Wrév = nRT v2 v1 dV V = nRTln V2 V1 = nRTln P1 P2

Isochoric Transformation (ΔV =0)

• An elementary variation dU = δQ v + δω.
• δW = -Pext dV = 0 (since dV = 0)
• It reduces as follows:
• dU = δQv.
• ∆U = Qv = Cv ∆T = n cỷ ∆Τ
• Thermal energy Q exchanged depends on the final and initial state and system variation.

Isobaric Transformation (ΔP = 0): Enthalpy

• Occurs at constant atmospheric pressure.
• The system exchanges heat and work with its surroundings.
• For an elementary transformation:
• dU = δQp + δW.
• δQp = heat exchanged at constant P.
• δW = -Pext dV = −P dV ......(P = P ext = P atmosphérique).
• dU = δQ P - P dV.
• For an evolution between two states 1 and 2:
• ΔU = Qp - PΔV.
• U2-U1 = QP - PV2 + PV1.
• Qp = (U 2+ PV2) — (U₁ + PV₁).
• Qp = Δ(U + PV).
• Quantity of heat exchanged under constant pressure equals the variation of a new function: H = U + PV.
• Enthalpy (H) is a state function depending on U, P, and V, and is an extensive function.
• It is written for an elementary transformation: 8Qp = dH = dU + d(PV) = dU + P dV, because VdP = 0 (P = cte).

Finite Transformation

• Qp = ΔH = ΔU + PΔV.
• This relationship applies to reversible and irreversible transformations.
• Joule's second law states that variables adapted to H are T and p with a differential written as: dH = (∂H/∂P)T dP + (∂H/∂T)p dT.
• At constant temperature, the enthalpy variation is zero.
• Enthalpy depends only on temperature.

Joule's Second Law

• dH = (∂H/∂T) dT=CdT =ncpdT
• ΔH = ∫ CP dT = ∫ n cp dT = n cP ΔT.
• The transformation undergone by the ideal gas: ΔH = CPAT = ncpAT

Relationship between ΔH and ΔU for an Ideal Gas

• According to Joule's first law: ΔU = Cv ΔT = ncv ΔT

• According to Joule's second law: ΔH = CP ΔT = ncp ΔT

• Mayer equation: During an elementary transformation: dH = dU + PdV and PdV = nRdT.

• Hence: dH = dU + nRdT.

• ncpdT = ncdT + nRdT

• ncpΔT = ncv ΔT + nRΔT.

The Adiabatic Constant y

Cp is always Greater than Cv, and the coefficient y is always greater than 1 the adiabatic constant:

Values = Cp Cv: (1) for a monatomic gas : γ = 3 5 ≈ 1.67

(2) for a diatomic gas with rigid molecules: γ = 5 7≈ 1.4

(3) for a diatomic gas with non-rigid molecules: γ = 7 9 ≈ 1.28 solving the equations:

Reversible Adiabatic Transformation of an Ideal Gas (Laplace's Law)

• During an adiabatic transformation, the system cannot exchange heat: dU = δW + δQ = δW
• δW1→2 = dU = n cv dT , therefore W1→2 = cp dT = ncv ΔT

Gas Laws

• dH = dU + d(PV)=dU+PdV+VdP
• dH = -Pext dV + P dV + VdP ⇒dH = (P − P ext )dV + VdP
• The transformation is mechanically reversible with pressure balance:

dU = -PdV; dH = VdP

• dU = -PdV = CvdT; dH = VdP = C pdT
• Ratio of these two equations =VdP / -PdV
• -dP / PdV integrated into In PV = cte
• Laplace's law, to be applied by checking the three hypotheses: PV = cte
• Adiabatics as a function of temperature and volume as well as temperature versus pressure: TVP-1 = cte
• TYP1-Y= cte
• Only is valid for ideal gas transformation

Work Calculation For A Reversible Adiabatic Transformation

Calculation of Energy and Enthalpy

Clapeyron Diagram

• Clapeyron Diagram or PV diagram represents a thermodynamic system's pressure as a function to follow the transformation's evolution.
• Diagram can only be drawn when the pressure is set in the system.

Thermal Capacity

• Different substances are affected to magnitudes with addition of heat.
• Constant of proportionality between object heat loss or gained Q to the change in temperature.