MAE 113: Introduction to Propulsion

June 23, 2024

The Scope of Thermodynamics

▶ Study of Thermodynamics begins with Four Laws—The Zeroth, First, Second, and the Third Law of Thermodynamics.

▶ The laws are facts of experience.

▶ No mathematical proof exists.

▶ Validity lies in the absence of contrary experience.

▶ From these laws one can derive a network of equations that have application in physical sciences and engineering.

System & Surroundings

▶ System

▶ Is the topic of attention.

▶ Could be a furnace, automobile engine, jet engine or rockets.

▶ Surroundings

▶ The rest of the universe that is not included in the system.

▶ Observations of the system are made from the surroundings

State Functions and Path Functions

▶ State Functions

▶ Properties that do not depend on the past history of the substance nor on the means by which it reaches a given state. Examples pressure (p), temperature (T).

▶ Path functions

▶ Quantities such as “heat” and “work” are not properties, they account for energy changes that occur in the surroundings (not the system)

State Functions

▶ The Laws of Thermodynamics are used to define the changes in the intensive state functions, pressure, p, and temperature, T, and the extensive state functions internal energy, U, and entropy, S.

▶ The extensive state functions enthalpy, H, Gibbs free energy, G, and Helmholtz free energy, A are defined for convenience.

First Law of Thermodynamics

▶ The first law of thermodynamics is an extension of the law of conservation of energy, that energy can be neither created nor destroyed.

▶ It gives clarify the meaning of the elusive concept of “energy”.

▶ Fundamental Statement of the First Law of Thermodynamics is that there exists an extensive function called internal energy, U, having the the property that for an isolated system

The internal energy of an isolated system is constant.

▶ As a consequence it follows that for a closed system

∆U = Q + W        (1)

where ∆U is the change in internal energy of the system and is equal to the heat added Q to the system and W the work done on the system. Note that Q and W are path functions.

▶ For small changes eqn (1) is

δU = δQ + δW          (2)

Second Law of Thermodynamics

▶ In the classical formulation, the second law of thermodynamics states that there exist an absolute scale for the temperature T and an extensive function entropy S, such that for an infinitesimal process in a closed system

TδSsystem≥δQ          (3)

where the equality holds for reversible processes and the inequality is valid for natural processes. Here δQ is heat transferred to the system from the surroundings.

▶ As far as the surroundings are concerned the heat transfer is reversible, Hence δSsurroundings = −δQ/T.

▶ Hence it follows that

δSuniverse = δSsurroundings + δSsystem≥0

Fundamental Equations of Chemical Thermodynamics

From the first and second law of thermodynamics

δU = δQ + δW

TδS ≥ δQ

For a reversible process

δWrev = −pδV

δQrev = TδS

Hence

δU = TδS − pδV

Fundamental Equations of Chemical Thermodynamics

The equation

δU = TδS − pδV        (4)

is valid for all processes because it relates state functions. Define for convenience, the extensive functions enthalpy, H, Gibbs Free Energy G, and Helmholtz Free Energy, A as

H ≡ U + pV

G ≡ H − TS

A ≡ U − TS             (5)

It follows that

δH = TδS + Vδp      (6)

δG = −SδT + Vδp    (7)

δA = −SδT − pδV    (8)

Equations (4), (6), (7), (8) are called the fundamental equations of chemical thermodynamics.

Notation

▶ In the subsequent development, V, U, S, H, G, A, will be the volume, internal energy, entropy, enthalpy, Gibbs free energy and Helmholtz free energy per mol respectively.

▶ Q and W will represent heat addition to the system and work done on the system per mol.

Heat Capacity

Introduce the following definitions for the heat capacity, Cp, and CV

Cp≡ (∂T/∂H)p; CV≡ (∂T/∂U)V       (9)

▶ For a constant volume process from Eq. (9) δU = CVδT. If only work due to expansion or compression is considered, for constant V, δW = 0, as a consequence δU = δQ. Hence CVδT = δQ .

▶ For a constant pressure process from Eq. (9) δH = CpδT. Also δH = δU + pδV + Vδp = δQ + δW + pδV + Vδp. For a reversible process δWrev = −pδV. Hence for a constant pressure reversible process δH = CpδT = δQ

Criteria for Chemical Equilibrium

Consider a closed system that is not in chemical equilibrium and in which chemical reactions are taking place.

▶ From the statement of the second law of thermodynamics

∆Ssys + ∆Ssurr ≥ 0

▶ At equilibrium ∆Ssys + ∆Ssurr = 0. Hence the fundamental criteria for chemical equilibrium is that the total entropy of the universe which is the sum of the entropy of the system and entropy of the surroundings must be a maximum.

▶ If the system is isolated,

▶ it does not interact with the surroundings. Hence ∆Ssurr = 0, ∆Ssys ≥ 0 as chemical reactions take place, and at chemical equilibrium ∆Ssys = 0

▶ Hence for an isolated system at chemical equilibrium the entropy of the system, Ssys = maximum

Criteria for Chemical Equilibrium, Constant temperature and Pressure

Consider a closed system that is in mechanical and thermal equilibrium but not in chemical equilibrium and in which chemical reactions are taking place at constant temperature and pressure. It will be assumed

T = Tsys = Tsurr; p = psys = psurr

As chemical reaction proceed

∆Ssys + ∆Ssurr ≥ 0

Also ∆Ssurr = −∆Q/T where ∆Q is the heat added to the system. At constant pressure and temperature it was shown that ∆Hsys = ∆Qsys. Hence ∆Ssys − ∆Hsys/T≥ 0. It follows that

∆ (Hsys − TSsys) = ∆Gsys ≤ 0

At constant pressure and temperature at chemical equilibrium Gibbs Free Energy is a minimum