\[\Delta S = \int_{T_1}^{T_2}\frac{C_p}{T}dT = C_p\ln\frac{T_2}{T_1}\ (\text{恒压，} C_p\text{ 恒定}),\qquad \Delta S = nC_{V,m}\ln\frac{T_2}{T_1} + nR\ln\frac{V_2}{V_1}\]\[\Delta S = \int_{T_1}^{T_2}\frac{C_p}{T}dT = C_p\ln\frac{T_2}{T_1}\ (\text{const. p ,} C_p\text{const.}),\qquad \Delta S = nC_{V,m}\ln\frac{T_2}{T_1} + nR\ln\frac{V_2}{V_1}\]
含义Meaning
加热使熵增加，增量与温度的对数成正比。理想气体同时改变温度和体积时两项相加。
Heating raises entropy in proportion to the log of temperature; for an ideal gas changing both T and V the two contributions add.
符号说明Symbols
| 符号Symbol | 含义 Meaning | 单位 Unit |
|---|---|---|
| \(C_p\) | 定压热容heat capacity at constant pressure | J·K⁻¹ |
推导Derivation
恒压可逆加热 \(\delta q_\text{rev} = C_pdT\)，代入 dS = δq_rev/T 积分。
Insert δq_rev = CpdT and integrate.
理想气体任意过程：\(dS = (dU + p\,dV)/T = nC_V\,dT/T + nR\,dV/V\)。
For an ideal gas dS = nCv dT/T + nR dV/V.
适用条件与局限Assumptions & limitations
热容随温度变化时需用实际 Cp(T) 积分；跨越相变要加相变熵。
Use Cp(T) if it varies; add transition entropies.
例题Worked example
1 mol 水从 0 °C 加热到 100 °C：ΔS = 75.3×ln(373/273) = 23.5 J/K。
Heating 1 mol of water 0 → 100 °C: ΔS = 23.5 J/K.