Chapter 4 Summary: First Law of Thermodynamics and Ideal Gases

Operating Principles and Energy Exchanges

  • The operating principle of thermal machines is founded on energy exchanges between a working fluid and two distinct parts of the external environment maintained at different temperatures.

  • These energy exchanges occur in two forms:

    • Work: represented by WW.

    • Heat: represented by QQ.

  • Energy exchanges modify the total energy content of the working fluid (EtotE_{tot}). The expression for total energy is:

    • Etot=Emacro+U=Ek,macro+Ep,macro+UE_{tot} = E_{macro} + U = E_{k,macro} + E_{p,macro} + U

  • Work and heat exchanged between the working fluid and the external environment alter this energy content:

    • ΔEtot=ΔEmacro+ΔU=ΔEk,macro+ΔEp,macro+ΔU=W+Q\Delta E_{tot} = \Delta E_{macro} + \Delta U = \Delta E_{k,macro} + \Delta E_{p,macro} + \Delta U = W + Q

  • For elementary exchanges and changes, the relationship is expressed as:

    • dEtot=dEmacro+dU=δW+δQdE_{tot} = dE_{macro} + dU = \delta W + \delta Q

The First Law of Thermodynamics

  • The law simplifies under specific physical conditions:

    • If a thermodynamic system is at rest or moving at a constant speed, its macroscopic kinetic energy does not change: ΔEk,macro=0\Delta E_{k,macro} = 0.

    • If the altitude of the system does not vary and there are no other interactions involving potential energy: ΔEp,macro=0\Delta E_{p,macro} = 0.

  • Under these established conditions, the energy exchanges in the form of work (WW) and heat (QQ) determine the change in the system’s internal energy (UU):

    • ΔU=W+Q\Delta U = W + Q

  • In differential form for elementary exchanges:

    • dU=δW+δQdU = \delta W + \delta Q

Internal Energy and Heat Capacity of Ideal Gases

  • Fundamental principle for the internal energy of an ideal gas: each 'degree of freedom' (#dof) contributes 12nRT\frac{1}{2} nRT to the total internal energy.

  • General formula for internal energy:

    • U=#dof2nRTU = \frac{\#dof}{2} nRT

  • For a closed system, such as the working fluid of a thermal machine, the elementary change in internal energy is solely related to an elementary change in temperature:

    • dU=#dof2nR dTdU = \frac{\#dof}{2} nR \, dT

Heat Capacity at Constant Volume (CVC_V)

  • In an isochoric process (constant volume), the elementary energy exchanges are defined as:

    • δW=0\delta W = 0

    • δQ=CV dT\delta Q = C_V \, dT

  • Using the first law (dU=δW+δQdU = \delta W + \delta Q), we derive the heat capacity at constant volume for an ideal gas:

    • #dof2nR dT=CV dT\frac{\#dof}{2} nR \, dT = C_V \, dT

    • CV=#dof2nRC_V = \frac{\#dof}{2} nR

  • Consequently, the change of internal energy for a closed system of ideal gas is always:

    • dU=CV dTdU = C_V \, dT

Enthalpy and Heat Capacity at Constant Pressure (CPC_P)

  • Enthalpy (HH) is defined by the relation:

    • H=U+PVH = U + PV

  • The differential of enthalpy is:

    • dH=dU+P dV+V dPdH = dU + P \, dV + V \, dP

  • Substituting the first law (dU=δW+δQdU = \delta W + \delta Q) into the enthalpy differential:

    • dH=δW+δQ+P dV+V dPdH = \delta W + \delta Q + P \, dV + V \, dP

  • For reversible processes where δW=−P dV\delta W = -P \, dV, this simplifies to:

    • dH=δQ+V dPdH = \delta Q + V \, dP

  • For an ideal gas, Enthalpy can be expressed as:

    • H=U+nRTH = U + nRT

    • In terms of degrees of freedom: H=(#dof2+1)nRTH = (\frac{\#dof}{2} + 1) nRT

  • The elementary change in enthalpy for a closed system is:

    • dH=(#dof2+1)nR dTdH = (\frac{\#dof}{2} + 1) nR \, dT

  • In an isobaric process (dP=0dP = 0), the heat exchange is δQ=CP dT\delta Q = C_P \, dT, leading to:

    • dH=CP dTdH = C_P \, dT

    • CP=(#dof2+1)nRC_P = (\frac{\#dof}{2} + 1) nR

Heat Capacity Ratio and Thermodynamic Relations

  • The heat capacity ratio (γ\gamma) is defined as:

    • γ=CPCV\gamma = \frac{C_P}{C_V}

  • For an ideal gas closed system, γ\gamma relates to the degrees of freedom:

    • γ=#dof2+1#dof2=1+2#dof\gamma = \frac{\frac{\#dof}{2} + 1}{\frac{\#dof}{2}} = 1 + \frac{2}{\#dof}

  • From this, we can derive ratios for the degrees of freedom:

    • #dof2=1γ−1\frac{\#dof}{2} = \frac{1}{\gamma - 1}

    • #dof2+1=γγ−1\frac{\#dof}{2} + 1 = \frac{\gamma}{\gamma - 1}

  • Heat capacities expressed via the heat capacity ratio:

    • CV=nRγ−1C_V = \frac{nR}{\gamma - 1}

    • CP=γnRγ−1C_P = \frac{\gamma nR}{\gamma - 1}

Summary of Gas Types for Closed Systems

  • Monoatomic Gas:

    • CV=32nRC_V = \frac{3}{2} nR

    • CP=52nRC_P = \frac{5}{2} nR

    • γ=53\gamma = \frac{5}{3}

  • Diatomic Gas:

    • CV=52nRC_V = \frac{5}{2} nR

    • CP=72nRC_P = \frac{7}{2} nR

    • γ=75\gamma = \frac{7}{5}

  • Triatomic Gas:

    • CV=72nRC_V = \frac{7}{2} nR

    • CP=92nRC_P = \frac{9}{2} nR

    • γ=97\gamma = \frac{9}{7}

Change in Internal Energy across Processes

  • For a process moving from state AA to state BB, the total change in internal energy is:

    • ΔUAB=CVΔTAB=nRγ−1ΔTAB=PBVB−PAVAγ−1\Delta U_{AB} = C_V \Delta T_{AB} = \frac{nR}{\gamma - 1} \Delta T_{AB} = \frac{P_B V_B - P_A V_A}{\gamma - 1}

  • Specific process simplifications:

    • Isothermal process: ΔUAB=0\Delta U_{AB} = 0

    • Isobaric process: ΔUAB=PΔVABγ−1\Delta U_{AB} = \frac{P \Delta V_{AB}}{\gamma - 1}

    • Isochoric process: ΔUAB=VΔPABγ−1\Delta U_{AB} = \frac{V \Delta P_{AB}}{\gamma - 1}

Reversible Adiabatic Processes of Closed Ideal Gas Systems

  • An adiabatic process is defined by zero heat exchange: δQ=0\delta Q = 0.

  • According to the first law, dU=δWdU = \delta W.

  • For a reversible process where dU=CV dTdU = C_V \, dT and δW=−P dV\delta W = -P \, dV, we obtain:

    • CV dT=−P dVC_V \, dT = -P \, dV

  • Using the ideal gas equation (P=nRTVP = \frac{nRT}{V}) and the definition CV=#ddl nR2C_V = \frac{\#ddl \, nR}{2}, the differential equation becomes:

    • #dof2dTT=−dVV\frac{\#dof}{2} \frac{dT}{T} = -\frac{dV}{V}

  • Integrating from state AA to state BB yields:

    • (TBTA)#dof2=VAVB(\frac{T_B}{T_A})^{\frac{\#dof}{2}} = \frac{V_A}{V_B} which is equivalent to TBTA=(VAVB)γ−1\frac{T_B}{T_A} = (\frac{V_A}{V_B})^{\gamma - 1}

  • State variable relationships during reversible adiabatic processes:

    • TVγ−1=const.T V^{\gamma - 1} = \text{const.}

    • T∝V1−γT \propto V^{1 - \gamma}

    • PVγ=const.P V^{\gamma} = \text{const.}

    • TγP1−γ=const.T^{\gamma} P^{1 - \gamma} = \text{const.}

  • Graphic Interpretation:

    • The proportionality between pressure and volume is P∝V−γP \propto V^{-\gamma}, where \gamma > 1.

    • In a pressure-volume (PVPV) diagram, an adiabatic process is represented by a hyperbola that is steeper than an isothermal process (P∝V−1P \propto V^{-1}).

Heat of a Reversible Isothermal Process

  • For a closed ideal gas system undergoing a reversible isothermal process, ΔUAB=0\Delta U_{AB} = 0.

  • The first law dictates that the heat exchange is the negative of the work exchange:

    • QAB=−WABQ_{AB} = -W_{AB}

  • The heat can be calculated using several equivalent expressions:

    • QAB=nRTln⁡(VBVA)Q_{AB} = nRT \ln(\frac{V_B}{V_A})

    • QAB=PAVAln⁡(VBVA)Q_{AB} = P_A V_A \ln(\frac{V_B}{V_A})

    • QAB=PBVBln⁡(VBVA)Q_{AB} = P_B V_B \ln(\frac{V_B}{V_A})

    • QAB=PBVBln⁡(PAPB)Q_{AB} = P_B V_B \ln(\frac{P_A}{P_B})

    • QAB=PAVAln⁡(PAPB)Q_{AB} = P_A V_A \ln(\frac{P_A}{P_B})

    • QAB=nRTln⁡(PAPB)Q_{AB} = nRT \ln(\frac{P_A}{P_B})