Thermodynamics: Energy and Systems

Thermodynamics

• Thermodynamics explores the relationship among heat, mechanical work, and internal energy.
• The first law of thermodynamics: the change in internal energy (ΔU) equals the heat added (Q) minus the work done (W), expressed as ΔU = Q - W.

Internal Energy (U)

• Internal energy (U) is directly proportional to temperature.
• Heat flowing into the system results in a positive Q, increasing internal energy.
• Heat flowing out of the system results in a negative Q, decreasing internal energy.

Work (W)

• Work done by the system is positive, which decreases the internal energy.
• Work done on the system is negative, increasing the internal energy.

Open, Closed, and Isolated Systems

• Open System: Allows the exchange of both matter and energy with the surroundings.
• Closed System: Allows energy exchange but not matter exchange.
• Isolated System: Prevents exchange of both energy and matter, maintaining constant internal energy (ΔU = 0).

Thermodynamic Processes

• Isochoric Process: Constant volume (ΔV = 0), resulting in zero work done (W = 0).
• Isobaric Process: Constant pressure (ΔP = 0).
• Isothermal Process: Constant temperature (ΔT = 0), leading to no change in internal energy (ΔU = 0).
• Adiabatic Process: No heat exchange occurs (Q = 0).

Isochoric Process Equations

• W = 0 because volume remains constant.
• ΔU = Q, indicating the change in internal energy equals heat flow.
• Q = n * CV * ΔT, used for heat flow calculation.
  • n = number of moles.
  • CV = molar heat capacity at constant volume for an ideal gas.
  • ΔT = change in temperature, measured in Kelvin or Celsius.
  • P1 / T1 = P2 / T2 is a notable equation.

Isobaric Process Equations

• W = P * ΔV, determines work using pressure and volume change.
• W = n * R * ΔT, alternative calculation for work.
• Q = n * CP * ΔT, is used to calculate heat flow.
  • CP = molar heat capacity at constant pressure.
• ΔU = Q - W, calculates the change in internal energy.
• ΔU = n * CV * ΔT, provides another method to calculate the change in internal energy.
• V1 / T1 = V2 / T2 is a useful equation.

Isothermal Process Equations

• ΔU = 0, the change in internal energy is always zero during the process.
• Q = W, heat flow equals the work done.
• W = n * R * T * ln(Vf / Vi), work calculation using volume.
  • Vf = final volume.
  • Vi = initial volume.
• W = n * R * T * ln(Pi / Pf), work calculation using pressure.
  • Pi = initial pressure.
  • Pf = final pressure.
• P1 * V1 = P2 * V2, this is a key equation.

Adiabatic Process Equations

• Q = 0, signifying no heat flow.
• ΔU = -W, change in internal energy equals negative work done.
• ΔU = n * CV * ΔT calculates change in internal energy.
• W = -n * CV * ΔT calculates work done.
• CV = (3/2) * R for monoatomic gas

Molar Heat Capacity

• CP = (5/2) * R, describes molar heat capacity at constant pressure for a monoatomic ideal gas.
• For diatomic gases, CV ≈ (5/2) * R and CP ≈ (7/2) * R.
• For polyatomic gases, CV ≈ (7/2) * R and CP ≈ (9/2) * R, but these values are approximate.

Ideal Gas Law

• PV = nRT (Ideal Gas Law)
• Value of R = 8.3145 J/(mol·K)
• Value of R = 0.08206 L⋅atm/(mol·K)

Conversions

• 1 Pascal * 1 cubic meter = 1 Joule
• 1 liter * 1 atm = 101.3 Joules
• 1 atm = 1.013 * 10^5 Pascals
• 1 cubic meter = 1,000 liters

PV Diagrams

• The area under a PV diagram indicates the work done on or by the gas.

Cyclic Process

• Returns a system to its initial state.
• Total work is the area inside the cycle on a PV diagram.
• ΔU = 0, therefore Q = W.
• Efficiency (η) = (Output Work / Heat Input) * 100%

Not State Function

• Work is not a state function

Additional Equations for Isobaric Process

• Q = P * CP * ΔV / R (Heat equation)
• P * ΔV = NR * ΔT (Ideal Gas Law)

Gamma Ratio

• γ (gamma) = CP / CV (ratio of molar heat capacities)
• CV = R / (γ - 1)
• CP = R * γ / (γ - 1)

Adiabatic Expansion/Compression

• P1 * V1^γ = P2 * V2^γ
• T1 * V1^(γ-1) = T2 * V2^(γ-1)

Isobaric Process

• Pressure remains constant during the process.
• Volume increases as temperature increases.
• Governed by Charles's Law: V1 / T1 = V2 / T2.

Cyclic Process

• The work done by the gas is equivalent to the enclosed area within the cycle on a PV diagram.
• In a rectangular cycle, work = change in pressure × change in volume, involving both isobaric and isochoric processes.
• With changing pressure at a constant rate, work done equals average pressure × volume change.
• Average pressure formula: (Initial Pressure + Final Pressure) / 2.
• Work formula: [(Initial Pressure + Final Pressure) / 2] × Change in Volume.
• Clockwise direction indicates positive work.
• Counterclockwise direction indicates negative work.
• Net heat flow is equal to work done (Q = W).
• Change in internal energy (ΔU) equals zero.
• Net heat flow is the difference between heat entering (Qh) and heat leaving (Ql).
• Qh is the total heat absorbed.
• Ql is the total energy released.
• Work is the difference between total heat absorbed and total energy released (W = Qh - Ql).
• Efficiency Calculation: Work performed / Total heat input (W / Qh), often expressed as a percentage.

Isothermal Process

• Temperature is constant (ΔT = 0).
• Displayed as an isothermal graph on a PV diagram.
• Work equation: W = nRT * ln(V_final / V_initial)
• Work equation: W = nRT * ln(P_final / P_initial)
• The Change in internal energy (ΔU) is zero.
• ΔU = nCvΔT, where ΔU is zero because ΔT is zero.
• Since ΔU = Q - W = 0, Q = W.
• Volume and pressure are inversely proportional.
• This process follows Boyle's Law: P1V1 = P2V2

Adiabatic Process

• There is virtually no heat exchange with the surroundings (Q ≈ 0).
• Change in internal energy matches work done: ΔU = W
• ΔU = nCvΔT applies to all processes.
• Therefore, W = -nCvΔT
• Cv = R / (γ - 1), with R as 8.3145 J/(mol·K)
• Cp = R * γ / (γ - 1)
• γ represents the ratio of Cp to Cv.
• P1V1^γ = P2V2^γ
• T1V1^(γ-1) = T2V2^(γ-1)
• W = (-Cv / R) * (P_final * V_final - P_initial * V_initial)
• Consequently, W = -1 / (γ - 1) * (P_final * V_final - P_initial * V_initial)