Entropy describes the tendency for systems to become disordered over time. The amount of [[Heat|thermal energy]] that can be converted into [[Work|work]] will decrease over time, entropy is a measure of how much thermal energy is unavailable. Entropy is measured in $J/K$, and is a property of [[Substance|substances]] that can be found in tables. It can be shown that the integral $\oint\dfrac{\partial Q}{T}$ for a reversible system is [[Path Dependence|independent of path]], proving that it is a property based on the [[State|state]] of a [[Substance|substance]]. In fact, this property is called entropy, $S$, and has the following relation: $S_2−S_1=∫_1^2\left(\dfrac{\partial Q}{T}\right)_{rev}$ Similar to [[Specific Volume and Density|specific volume]] and specific work we can define specific entropy as: $s=\dfrac{S}{m}$ It is also evident that heat transfer during a process can be defined by entropy. $Q_{1\to2}=\int_1^2TdS$ Therefore, just as specific work can be thought of as the area under a [[Tv and Pv Diagrams|P-v curve]], heat can be thought of as the area under a T-s curve. Finally, using the first law we can relate all substance properties to one another with whats called the Gibbs equation: $TdS=dU+PdV$ $TdS=dH−VdP$ ## Solids and Incompressible Liquids: $d\nu=0$ $du=CdT$ $Tds=CdT$ $s_2−s_1=C\ln⁡ \left(\dfrac{T_2}{T_1}\right)$ ## Compressible Liquids: $s\approx s_f$ ## Saturated Mixtures: $s=s_f+xs_{fg}$ ## Superheated Vapours: Look up in relevant substance tables. ## Ideal Gases $du=C_{v0}dT$ $dh=C_{p0}dT$ $P\nu=RT$ $Tds=C_{v0}dT+Pd\nu$ $s_2−s_1=C_{v0}\ln\left(\dfrac{T_2}{T_1}\right) + R \ln\left(\dfrac{\nu_2}{\nu_1}\right)$ $Tds=C_{p0}dT−\nu dP$ $s_2−s_1=C_{p0}\ln \left(\dfrac{T_2}{T_1}\right)−R\ln⁡\left(\dfrac{P_2}{P_1}\right)$ | Polytropic Exponent | Condition | Result | | ------------------------------ | ---------- | --------------------- | | $n=0$ | Isobaric | $P=\text{constant}$ | | $n=1$ | Isothermal | $T=\text{constant}$ | | $n=k=\dfrac{C_{p0}}{C_{v0}}​​$ | Isentropic | $s=\text{constant}$ | | $n=\infty$ | Isochoric | $\nu=\text{constant}$ | ## Overall Entropy Change The entropy change for an [[Reversibility|irreversible]] process must be larger than that for a reversible process. Therefore, it is clearly true that the following holds for an irreversible process: $S_2−S_1\ge\int_1^2\dfrac{\partial Q}{T}​$ As such, we will define a term called entropy generation, $S_{gen}$​. Entropy generation represents the entropy created by the factors leading to irreversibility, such as [[Friction|friction]] and [[Mixtures|mixing]]. $S_2−S_1=\int_1^2\dfrac{\partial Q}{T}+S_{gen}$ For a [[Control Mass|control mass]], denoted with subscript $cm$, the entropy generation can be defined as: $S_{gen}=m_{cm}(s_2−s_1)_{cm}−\dfrac{Q_{1\to2}}{T_{surr}}$