\[pV^\gamma = \text{常数},\qquad TV^{\gamma-1} = \text{常数},\qquad \gamma = \frac{C_{p,m}}{C_{V,m}},\qquad w = \Delta U = nC_{V,m}(T_2 - T_1)\]\[pV^\gamma = \text{const.},\qquad TV^{\gamma-1} = \text{const.},\qquad \gamma = \frac{C_{p,m}}{C_{V,m}},\qquad w = \Delta U = nC_{V,m}(T_2 - T_1)\]
含义Meaning
与外界无热交换（q = 0）时，膨胀做功全部来自内能，气体冷却。绝热线比等温线更陡。
With no heat exchange the work comes entirely from internal energy, so an expanding gas cools; adiabats are steeper than isotherms.
符号说明Symbols
| 符号Symbol | 含义 Meaning | 单位 Unit |
|---|---|---|
| \(\gamma\) | 热容比（单原子 5/3，双原子 7/5）heat-capacity ratio (5/3 monatomic, 7/5 diatomic) | 1 |
推导Derivation
q = 0：\(dU = \delta w\)，即 \(nC_{V,m}dT = -p\,dV = -nRT\,dV/V\)。
With q = 0, \(nC_VdT = -nRT\,dV/V\).
分离变量积分：\(C_{V,m}\ln(T_2/T_1) = -R\ln(V_2/V_1)\)，用 \(R = C_p - C_V\) 得 \(TV^{\gamma-1}\) = 常数。
Integrate and use R = Cp − Cv.
代入 \(T = pV/nR\) 得 \(pV^\gamma\) = 常数。
Substitute T = pV/nR.
适用条件与局限Assumptions & limitations
可逆、理想气体、γ 恒定；不可逆绝热（如向真空膨胀）不满足。
Reversible, ideal gas, constant γ.
例题Worked example
柴油机压缩比 18：空气（γ = 1.4）从 300 K 绝热压缩到 T = 300×18^0.4 = 953 K，足以点燃柴油。
Diesel compression ratio 18 heats air from 300 K to 953 K.