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  • Entropy change of liquids and solids:

liquids and solids can be approximated as incompressiblesubstances since their specific volumes remain nearly constant during aprocess

$$d s=\frac{d u}{T}=\frac{c d T}{T}$$

The entropy change during a process:

Liquids, solids: $$\quad s_{2}-s_{1}=\int_{1}^{2} c(T) \frac{d T}{T} \cong c_{\mathrm{avg}} \ln \frac{T_{2}}{T_{1}} \quad(\mathrm{kJ} / \mathrm{kg} \cdot \mathrm{K})$$

  • Entropy change for isentropic processes of liquids and solids:

Isentropic:$$\quad s_{2}-s_{1}=c_{\mathrm{avg}} \ln \frac{T_{2}}{T_{1}}=0 \quad \longrightarrow \quad T_{2}=T_{1}$$

  • Entropy change of ideal gases:the differential entropy change of an ideal gasbecomes

$$d s=c_{v} \frac{d T}{T}+R \frac{d v}{v}$$

$$s_{2}-s_{1}=\int_{1}^{2} c_{v}(T) \frac{d T}{T}+R \ln \frac{v_{2}}{v_{1}}$$

$$s_{2}-s_{1}=\int_{1}^{2} c_{p}(T) \frac{d T}{T}-R \ln \frac{P_{2}}{P_{1}}$$

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