Ideal Gas Processes in Reversible Adiabatic and Isothermal Expansions | MIT OpenCourseWare

Summary

MIT OpenCourseWare presents a focused exploration of ideal gas processes to model mixing real materials in materials science. The lecture examines reversible adiabatic expansion, choosing entropy and pressure as independent variables, and derives how temperature relates to entropy and pressure through coefficients determined by a general strategy. It then analyzes isothermal expansion, derives the Gibbs free energy change for an ideal gas, and demonstrates how selecting fixed state variables simplifies computations. The session also covers adiabatic free expansion, entropy production, and the second law, emphasizing the power of thermodynamics to predict outcomes from initial and final states, even when the actual path is irreversible.

• Independent variables in adiabatic reversible processes and how fixed quantities become independent
• Derivation of temperature changes along S and P, and the role of ideal gas properties
• Isothermal expansion and the Gibbs free energy change for an ideal gas
• Entropy changes in adiabatic free expansion and the second law

Overview

The MIT OpenCourseWare lecture motivates studying ideal gases as a model for mixing real materials, a key material science motivation. It then walks through three core gas processes to develop general strategies for thermodynamic analysis and state-variable selection.

Reversible Adiabatic Expansion: Independent Variables and Strategy

The instructor asks what to use as independent variables for a reversible adiabatic expansion, noting that no heat is exchanged (adiabatic) and the process is reversible. A practical rule is introduced: if something is fixed, it is independent, or stagnant, and should be treated as an independent variable. For an adiabatic reversible process, entropy S is fixed (dS = 0), so S becomes an independent variable, and pressure P or volume V can be varied. The speaker demonstrates how to express the temperature T as a function of (S, P) by assuming a linear form for dT in terms of dS and dP, then uses the ideal-gas relation V = RT/P and a known CP to derive the coefficients M and N in an expression like dT = M dS + N dP. A key result identified is M CP/T = 1, which constrains M, and similarly N = T V α/CP for the case where T and P are taken as independent variables.
"The math is easy when we set the problem up right" - MIT Instructor

Ideal Gas Properties and Coefficients

With the ideal gas law, V = RT/P, and α = 1/T, along with CP = 5/2 R for a monatomic gas, the coefficients are used to obtain the relation between temperature, entropy, and pressure for an adiabatic process. This framework leads to the standard adiabatic relation P V^γ = constant and T V^(γ-1) = constant, where γ = CP/CV. The instructor notes several equivalent forms of the adiabatic law and implies a separable differential equation to solve for the temperature change during adiabatic expansion, reinforcing the versatility of the general strategy.

Isothermal Expansion and Gibbs Free Energy

Next, the session considers isothermal expansion, where temperature is fixed (dT = 0). Here, T is treated as fixed and P or V is varied. The objective is to obtain the Gibbs free energy change, ΔG, for a process taking the gas from P1 to P2. Using the differential form dG = -S dT + P dV and the isothermal condition, the calculation simplifies to dG = P dV, which for an ideal gas integrates to ΔG = RT ln(P2/P1) or equivalently RT ln(V1/V2). The instructor emphasizes how choosing independent variables can simplify the mathematics dramatically and connects this result to fundamental electrochemistry forms, such as the Nernst equation in other contexts.
"This is called the NERNST equation" - MIT Instructor

The discussion also notes that the Gibbs energy, when expressed with the molar volume, reveals the same structure across contexts, illustrating the universality of the logarithmic term in change-of-state expressions.

Adiabatic Free Expansion and Entropy

The lecture then covers adiabatic free expansion, where a gas expands into a vacuum inside a thermally insulated cylinder, performing no work (W = 0) and exchanging no heat (Q = 0). For an ideal gas this implies ΔU = 0 and ΔT = 0, so the gas does not change its temperature, but its entropy increases since the process is spontaneous and irreversible. To compute ΔS, the instructor advocates a path-independent approach: find a reversible path between the same initial and final states, such as an isothermal expansion, and compute ΔS along that path. This yields ΔS = ∫(C_p/T) dT - ∫(V α/β) dV with α and β from the ideal gas relations. For the ideal gas, the result simplifies to ΔS = R ln(V2/V1) > 0 for expansion into the vacuum, confirming the second law. The discussion highlights how the same endpoint can be reached by different paths, but state functions like S remain path-independent, while entropy production is associated with irreversibility. A contrast is drawn with the backward process, which would yield a negative ΔS and is not spontaneously realized.

"Spontaneous entropy has to always increase or stay the same" - MIT Instructor