定体, 定压, 定温, 绝热 Q E A 公式

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

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

• 求内能变化: ( riangle E = u C_V riangle t = ufrac{i}{2}R riangle 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. ( riangle v = 0, A = 0)

   • ( riangle E = u C_V riangle t = ufrac{i}{2}R riangle t = frac{i}{2}(P_2 V_2 - P_1 V_1))
   • (Q = riangle E)

2. ( riangle p = 0)

   • (A = P riangle v = u R riangle t)
   • ( riangle E = u C_V riangle t = ufrac{i}{2}R riangle t = frac{i}{2}P riangle v)
   • (Q = A + riangle E = u R riangle t + ufrac{i}{2}R riangle t = ufrac{2+i}{2}R riangle t = u C_P riangle t = frac{2+i}{2}P riangle v)

3. ( riangle t = 0, riangle E = 0)

   • (A = int_{v1}^{v2}PdV = int_{V_1}^{V_2}frac{ u RT}{v}dV = u RTint_{V_1}^{V_2}frac{dV}{v} = u RT(ln{V_2} - ln{V_1}) = u RT ln{frac{V_2}{V_1}} = u 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)

   • ( riangle E = u C_V riangle t = frac{i}{2}(P_2 V_2 - P_1 V_1))
   • (A = Q - riangle E = - riangle 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 + u C_Vdt = 0)
                 • (dQ= dA + dE)
                 • (dA = pdv)
                 • (dE = u C_Vdt)
               • (pdv + vdp = u Rdt)
                 • (pv = u Rt)
   • (v^{gamma-1}t = c2)
     • (v^{gamma - 1} u 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}}}{ u R})
       • (tp^{frac{1}{gamma}-1}=frac{c_{1}^{frac{1}{gamma}}}{ u R})
         • (frac{ u 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)