• 通用: \(PV = \frac{M}{M_{mol}}Rt = \nu Rt\)

    • \(M\): 质量
    • \(M_{mol}\): 1 mol物质的质量
    • \(\nu\): 几摩尔

  • 求内能变化: \(\triangle E = \nu C_V \triangle t = \nu\frac{i}{2}R\triangle t = \frac{i}{2}(P_2 V_2 - P_1 V_1)\)

    • \(C_v\): 等体摩尔热容
    • \(i\): 分子自由度
      • 单原子, \(i=3\)
      • 双原子, \(i=5\)
      • 多原子, \(i=6\)

  • 比容热比: \(\gamma = \frac{C_P}{C_V} = \frac{2+i}{i}\)

  1. \(\triangle v = 0, A = 0\)

    • \(\triangle E = \nu C_V \triangle t = \nu\frac{i}{2}R\triangle t = \frac{i}{2}(P_2 V_2 - P_1 V_1)\)
    • \(Q = \triangle E\)

  2. \(\triangle p = 0\)

    • \(A = P\triangle v = \nu R \triangle t\)
    • \(\triangle E = \nu C_V \triangle t = \nu\frac{i}{2}R\triangle t = \frac{i}{2}P\triangle v\)
    • \(Q = A + \triangle E = \nu R \triangle t + \nu\frac{i}{2}R\triangle t = \nu\frac{2+i}{2}R\triangle t = \nu C_P\triangle t = \frac{2+i}{2}P\triangle v\)

  3. \(\triangle t = 0, \triangle E = 0\)

    • \(A = \int_{v1}^{v2}PdV = \int_{V_1}^{V_2}\frac{\nu RT}{v}dV = \nu RT\int_{V_1}^{V_2}\frac{dV}{v} = \nu RT(ln{V_2} - ln{V_1}) = \nu RT ln{\frac{V_2}{V_1}} = \nu RT ln{\frac{P_1}{P_2}} = P_1V_1 ln{\frac{V_2}{V_1}} = P_2V_2 ln{\frac{P_1}{P_2}}\)
    • \(Q = A\)

  4. \(Q = 0\)

    • \(\triangle E = \nu C_V \triangle t = \frac{i}{2}(P_2 V_2 - P_1 V_1)\)
    • \(A = Q - \triangle E = - \triangle E\)
    • \(v^\gamma p=c_1\)
      • \(\gamma \ln{v} + \ln{p} = \ln{c_1}\)
        • \(\int \int \frac{\gamma}{v}dv + \frac{1}{p}dp = \ln{c_1}\)
          • \(\frac{\gamma}{v}dv + \frac{1}{p}dp = 0\)
            • \(\gamma pdv + vdp = 0\)
              • \(C_P = C_V + R\)
              • \(\gamma = \frac{C_P}{C_V}\)
              • \((C_V+R)pdv + C_Vvdp=0\)
                • \(pdv + \nu C_Vdt = 0\)
                  • \(dQ= dA + dE\)
                  • \(dA = pdv\)
                  • \(dE = \nu C_Vdt\)
                • \(pdv + vdp = \nu Rdt\)
                  • \(pv = \nu Rt\)
    • \(v^{\gamma-1}t = c2\)
      • \(v^{\gamma - 1}\nu Rt=c_1\)
        • \(v^{\gamma - 1}pv=c_1\)
          • \(v^\gamma p=c_1\)
    • \(t^{-\gamma} p^{\gamma-1} = c3\)
      • \(tp^{\frac{1-\gamma}{\gamma}}=\frac{c_{1}^{\frac{1}{\gamma}}}{\nu R}\)
        • \(tp^{\frac{1}{\gamma}-1}=\frac{c_{1}^{\frac{1}{\gamma}}}{\nu R}\)
          • \(\frac{\nu Rt}{p} p^{\frac{1}{\gamma}}=c_{1}^{\frac{1}{\gamma}}\)
            • \(v p^{\frac{1}{\gamma}}=c_{1}^{\frac{1}{\gamma}}\)
              • \(v^\gamma p=c_1\)