Thermodynamics: Entropy, Coefficients, and Ideal Gases

Posted on Feb 8, 2025 in Physics

Thermodynamics

Entropy

In a Carnot cycle, the integral of Q1/T1 from 0 to Q2/T2 equals 0, where T1 and T2 are the temperatures. This gives us Q1/T1 + Q2/T2 = 0. Considering ΔQ = at, we have at1/T1 + at2/T2 = 0, thus ∑ ΔQ/T = 0.

Now, for a cyclic transformation, ∑ ΔQ/T = ∫1a2 ΔQ/T + ∫2b1 ΔQ/T = 0. If we consider other ways of reasoning, ∫1N2 ΔQ/T, the ratio of heat transfer at different temperatures is a constant. This magnitude is called Entropy.

The third principle of thermodynamics states that the value of entropy at absolute zero temperature is zero. For irreversible and reversible processes, consider the heat transfer and indicate the variation of entropy. Given thermal machines, the variations of Q are known as variations of entropy. Having defined entropy, the variation in this concept of transfer is ΔSsystem > 0. Therefore, ΔSuniverse = ΔSsystem + ΔSenvironment > 0 in any real thermal transfer, meaning the universe’s entropy always increases.

Thermodynamic Coefficients

The thermodynamic state is defined by T, V, and P as independent variables. We can establish a relation of state, denoted as f(P, V, T) = 0. If we express one variable in terms of others, we can write dP = (∂P/∂V)dV + (∂P/∂T)dT, which can be rewritten as dP = (∂P/∂V)TdV + (∂P/∂T)VdT. The first term corresponds to constant T, and the second to constant V.

Specific heat is defined as C = 1/f (∂Q/∂T). Two common specific heats are:

  • Cp = (∂Q/∂T)P (at constant pressure)
  • Cv = (∂Q/∂T)V (at constant volume)

Where Cp > Cv. The ratio of molar heats is called the adiabatic index: γ = Cp/Cv > 1.

Other coefficients related to volume variations include:

  • Compressibility: K = -1/V (∂V/∂P)T
  • Expansion coefficient: β = 1/V (∂V/∂T)P

Matter exists in three states: solid, liquid, and gas. Transitions between these states involve heat transfer at constant temperature. These are called direct exchanges of heat when the system absorbs heat and indirect when heat is removed.

Gibbs Functions and Maxwell’s Equations

These functions have the same dimensions as energy. They help establish equilibrium conditions for chemical reactions and represent energetic variables in mechanics and heat.

1) Internal Energy (U)

dU = TdS – pdV. Considering dU = (∂U/∂S)VdS + (∂U/∂V)SdV, we have T = (∂U/∂S)V and -P = (∂U/∂V)S. Deriving, we get (∂T/∂V)S = (∂2U/∂S∂V) and -(∂P/∂S)V = (∂2U/∂S∂V). This leads to the Maxwell relation: (∂T/∂V)S = -(∂P/∂S)V

2) Helmholtz Free Energy (A)

A = U – TS. Deriving, dA = dU – TdS – SdT. Since dU – TdS = -pdV, we have dA = -SdT – pdV.

3) Gibbs Free Energy (G)

G = U – TS + pV. Deriving, dG = dU – TdS – SdT + pdV + VdP. Since dU – TdS = -pdV, we have dG = -SdT + VdP.

4) Enthalpy (H)

H = U + pV. Deriving, dH = dU + pdV + VdP. Since dU + pdV = TdS, we have dH = TdS + VdP.

Ideal Gas

An ideal gas satisfies the following laws:

Boyle’s Law

At constant temperature, the product of pressure and volume remains constant: (pV)T = constant.

Joule’s Law

The internal energy is only a function of temperature: U = f(T).