Thermodynamics - My Personal Dairy

Equation of State of Ideal Gas \[ A(T, V, \bar{n}) - A(T, V_R, \bar{n}) = \int_{V_R}^{V} \left( \frac{\partial A}{\partial V} \right)_{T, \bar{n}} dV = -\int_{V_R}^{V} P dV \tag{1} \] In Equation 1, the term \( V_R \) acts as an arbitrary reference volume, which corresponds to a specific but unknown reference Helmholtz free energy, \( A_R \) (where \( A(T, V_R, \bar{n}) = A_R \)). To see how this plays out mathematically, let's look at the most basic example using the ideal-gas equation of state: \[ P^{IG}(T, V, \bar{n}) = \frac{n R_g T}{V} \tag{2} \] By substituting this into the fundamental volume derivative of the Helmholtz energy and performing the integration, we get: \[ A^{IG}(T, V, \bar{n}) - A^{IG}(T, V_R, \bar{n}) = -\int_{V_R}^{V} P^{IG} dV \] \[ = -\int_{V_R}^{V} \frac{n R T}{V} dV = -n R_g T \ln\left(\frac{V}{V_R}\right) \tag{3} \] While Equation 3 allows us to calculate changes in Helmholtz free energy, it only provides the difference between states—we still do not kno...

Starting with: \[ dU = T dS - P dV + \sum_r \mu_r dn_r \tag{1} \] Notice that Equation 1 is now composed entirely of state functions. This introduces internal energy (\( U \)) as a thermodynamic potential, indicating that \( U \) is fundamentally a function of entropy, volume, and the number of moles. While it is mathematically possible to express internal energy using temperature and volume instead, the resulting equation loses its compact elegance—a natural mathematical framing referred to as a canonical relationship. However, applying Equation 1 directly requires functional models that relate variables like \( T \), \( P \), and \( \mu_i \) back to \( S \), \( V \), and \( \bar{n} \). Because such models are rare and typically impractical for standard chemical engineering applications, it is often more useful to mathematically eliminate entropy as an independent variable. We do this using an operation called a Legendre transformation: \[ dU - d(TS) = T dS - P dV + \sum_r \mu_r dn_r ...

First Law of Thermodynamics for a closed system, starting from the universal principle of conservation of energy . 1. Total Energy Conservation: Because the universe is isolated, the sum of all energy changes must equal zero: energy cannot be created or destroyed: $$\Delta(E_{kin} + E_{pot} + U)_{Sys} + \Delta(E_{kin} + E_{pot} + U)_{Surr} = 0\tag{1}$$  The universe is divided into two parts: the System ( Sys , the region of interest) and the Surroundings ( Surr , everything else) . The total energy of any body or region consists of: \(E_{kin}\): macroscopic kinetic energy (movement of the center of mass) \(E_{pot}\):  macroscopic potential energy (position in external fields, like gravity) . \(U\): microscopic internal energy (molecular vibrations, rotations, electronic states, intermolecular forces) 2. Practical Approximations & Boundary Transfers $$\Delta(U)_{Sys} = -\Delta(E_k + E_p + U)_{Surr} = Q + W\tag{2}$$  Neglecting macroscopic motion of the system: In ...