\[w = -nRT\ln\frac{V_2}{V_1} = -nRT\ln\frac{p_1}{p_2},\qquad \Delta U = 0,\quad q = -w,\quad \Delta S = nR\ln\frac{V_2}{V_1}\]
含义Meaning
理想气体内能只与温度有关，等温时 ΔU = 0，吸收的热全部转化为功。可逆膨胀做功最大。
An ideal gas’s internal energy depends only on T, so isothermally ΔU = 0 and all absorbed heat becomes work; the reversible expansion does the most work.
符号说明Symbols
| 符号Symbol | 含义 Meaning | 单位 Unit |
|---|---|---|
| \(V_1, V_2\) | 始、末体积initial and final volumes | m³ |
| \(\Delta S\) | 体系熵变entropy change of the system | J·K⁻¹ |
推导Derivation
可逆过程 \(p_\text{ex} = p = nRT/V\)。
Reversibly \(p_\text{ex} = p = nRT/V\).
\(w = -\int nRT\,dV/V = -nRT\ln(V_2/V_1)\)。
Integrate.
\(\Delta S = q_\text{rev}/T = nR\ln(V_2/V_1)\)。
ΔS = q_rev/T.
适用条件与局限Assumptions & limitations
只对理想气体；真实气体等温膨胀 ΔU ≠ 0（分子间作用）。
Ideal gases only.
例题Worked example
1 mol 理想气体 298 K 体积加倍：w = −8.314×298×ln2 = −1.72 kJ，ΔS = 5.76 J/K。
Doubling the volume of 1 mol at 298 K: w = −1.72 kJ, ΔS = 5.76 J/K.