波尔兹曼常数测定实验数据处理

波尔兹曼常数测定实验数据处理以T=27°C为例，说明由实验数据求得波尔兹曼常数过程：对于线方程TOC\o"1-5"\h\zU二aU+b 121先利用最小二乘法计算系数a与b，过程如下：令①丄U-U) 22i 2i计i=1其中U=aU+b 32计 1将3式代入2式得①二工U-aU-b) 42i 1ii=1根据最小二乘法，当①最小时，用①对a与b求偏导，并令偏导数等于0,可以求得a与b。于是有正U-aU-b)。① 2i 1i正U-aU-b)。① 2i 1i=—i=1—

da da迟U-aU-b)d① 2i 1idb db可以得到=_2工kJ-aU-bUL02i 1i 1ii=1=一2工U-aU-b)=02i 1ii=1Zuu=a工U)+b》U2i 1i 1i 1ii=1 i=1 i=1ZuaZu+nb5678ii=12i 1ii=1由以上两个式子可以求得2i 1i 2i 1ia=i=1 i=1 i=11i丿i2i 1i 2i 1ia=i=1 i=1 i=11i丿i=1 丿TOC\o"1-5"\h\zi=1 i=1inEULpUi=1 '10EU-aEU10了 2i 1ib= n下面代入T=27°C时的实验数据：本次所做实验所测得的U与U的值如表一所示，取n为12。12将所做实验数据代入以上9,10两式，可以求得a与b。其中EU=0.32+0.33+……+0.42+0.43=4.51ii=1EU=0.111+0.165+......+5.486+7.85=24.554i2i=1EUEU=4.5x24.554=110.4932i 1ii=1 i=1EUU)EUU)=0.32x0.111+0.33x0.165++0.42x5.486+0.43x7.85=10.067492i 1ii=1=0.32=0.322+0.332++0.422+0.432=1.70181ii=1'EUv2=(0.32+0.33+......+0.42+0.43》=20.25Ii=1 "丿于是nEUU)-nEUU)-EuEu2i1i 2i 1ia=— i—nEU)-'EUi=1 1丿1ii=112x10.06749-110.493=60.1212X1.7018-20.25XU-aZU2i 1i24.554—60.12x4.5b=i=1 i=1 = =—20.5n 12之后将a与b的值代入1式可以求得u*，即因变量的预期值,2U*=aU+b=60.12X0.32-20.5=-1.2111依次求出U*的值填入表一。)=(0.111+1.2616b=1.884*)，填入表一O21Z2i然后求出U—U/21依次求出U—UZ2i然后求出U—U/21依次求出U—U21可以求i1I5=1gIJ—U*VLi=12i 2i*21各函数拟合的标准差二J(1.884+0.68+……+0.056+6.242b二1.10521lnU二Ina+blnU,21于是可以用最小二乘法，最后求得的a与b分别为工lnU-bXlnUi 2i 1ilna=i=1 i=1n工CUlnU)一工n工CUlnU)一工lnU工lnU1 2i 1i 2i 1ib=—i=1 i=1 i=1n工CU）一［工lnU一、一"丿将实验数据代入方程可以求得各个值，分别填入表二。kT）其中ekT=a，所以U=bexpCiU）1i对于指数函数U二b2121对此式两边同时取对数得inU二aU+lnb，可以用最小二乘法，得21出a与b分别为nEKnUUL》lnU工Ua=—n、2a=—n、2工UHu工lnU-OLU2i将实验数据代入方程可求得各个值，分别填入表三。将其他各个温度时的实验数据按照以上方法，可以求出各自的»a,b的值，分别填入各个表中。由表，我们可以看到，将实验数据代入指数函数U二bexp(u.kT)21所得的标准差5最小，说明用指数函数拟合的最小，于是我们将各个温度下，用指数函数求得的a代入ekT=a，来求得的波尔兹曼常数k=e(aT)，其中e二1.602x10-19如下：T=33°C,T=43°C,所求得的k与公认值k=1.38054x10-23J/K相差很小。另外，我们还可以做出三种函数的拟合图，见附图，我们只做了T=27°C时的线性拟合图，可以看到，指数回归拟合的最好。

T=27°C线性回归乘幂回归指数回归U=bexp(eU/kT)U=2=aU+b1U2=aUb1U/V1U/V2U*/V2(U-U*)2/22V2u*/2V(U-U*)222/V2U*/V2(U-U*)2/V22210.320.111-1.2621.8840.1011.00E-040.1110.00E+0020.330.165-0.6600.6800.1584.90E-050.1641.00E-0630.340.246-0.0590.0930.2439.00E-060.2421.60E-0540.350.3560.5420.0350.3701.96E-040.3571.00E-0650.360.5191.1430.3900.5561.37E-030.5264.90E-0560.370.7791.7440.9300.8262.21E-030.7769.00E-0670.381.1372.3461.4601.2156.08E-031.1444.90E-0580.391.6822.9471.6001.7697.57E-031.6872.50E-0590.42.4883.5481.1242.5513.97E-032.4880.00E+00100.413.7354.1490.1703.6458.10E-033.6684.49E-03110.425.4865.2500.0565.1651.03E-015.4086.08E-03120.437.8505.3526.2427.2593.49E-017.9741.54E-0181.1050.2000.117a60.121.45E+0638.830b-20.514.4634.47E-07

T=33°C线性回归乘幂回归指数回归U=bexp(eU/kT)U=2aU+b1U2=aUb1U/V1U/V2U*/V2(U-U*)2/22V2U*/2V(U-U*)222/V2U*/V2(U-U*)2/V22210.30.115-1.1991.7270.1051.00E-040.1161.00E-0620.310.170-0.6180.6210.1643.60E-050.1700.00E+0030.320.251-0.0380.0830.2501.00E-060.2489.00E-0640.330.3700.5430.0300.3798.10E-050.3634.90E-0550.340.5421.1240.3380.5655.29E-040.5321.00E-0460.350.7771.7050.8600.8333.14E-030.7781.00E-0670.361.1372.2851.3181.2156.08E-031.1381.00E-0680.371.6492.8661.4811.7541.10E-021.6652.56E-0490.382.4453.4471.0032.5083.97E-032.4368.10E-05100.393.5934.0270.1893.5531.60E-033.5648.41E-04110.405.2284.6080.3844.9895.71E-025.2141.96E-04120.417.675.1896.1576.9475.23E-017.6281.76E-0381.0870.2250.0166a58.071.08E+0638.05b-18.6213.4071.28E-06

T=38°C线性回归乘幂回归U二aUb21指数回归U二bexp(eU/kT)U=2aU+b1U/V1U/V2U*/V2(U-U*)222/V2U*/V2(U-U*)2/22V2U*/V2(U-U*)2/V22210.290.128-1.21291.7980.1171.21E-040.1291.00E-0620.300.187-0.6130.6400.1804.90E-050.1881.00E-0630.310.273-0.01310.0820.2754.00E-060.2741.00E-0640.320.400.58680.0350.4121.44E-040.3984.00E-0650.330.5841.18670.3630.6117.29E-040.5783.60E-0560.340.8531.78660.8720.8961.85E-030.8401.69E-0470.351.2182.38651.3651.2986.40E-031.2204.00E-0680.361.7772.98641.4631.8617.06E-031.7749.00E-0690.372.5513.58631.0722.6438.46E-032.5787.29E-04100.383.7184.18620.2193.7180.00E+003.7478.41E-04110.395.4624.78610.4575.1847.73E-025.4462.56E-04120.47.885.3866.2207.1685.07E-017.9151.23E-0381.1020.2250.0166a59.998.88E+0537.39b-18.6112.7982.53E-06

T=43°C线性回归U=aU+b21乘幂回归U二aUb21指数回归U二bexp(eU]/kT)U/V1U/V2U*/V2(U-U*)222/V2U*/V2(U-U*)2/22V2U*/V2(U-U*)2/V22210.280.161-1.5312.8620.1481.69E-040.1649.00E-0620.290.237-0.7721.0180.2296.40E-050.2394.00E-0630.30.347-0.0130.1300.3494.00E-060.3470.00E+0040.310.5080.7460.0570.5252.89E-040.5041.60E-0550.320.741.5050.5850.7791.52E-030.7343.60E-0560.331.0782.2641.4061.1424.10E-031.0671.21E-0470.341.5713.0232.1071.6567.23E-031.5514.00E-0480.352.2873.7822.2342.3767.92E-032.2569.61E-0490.363.3224.5401.4843.3732.60E-033.2811.68E-03100.374.825.2990.2304.7445.78E-034.7722.30E-03110.386.8026.0580.5536.6113.65E-026.9391.88E-01120.399.9286.8179.6789.1356.29E-0110.0912.66E-0281.3650.2410.135a75.891.12E+0637.45b-22.7812.4464.58E-06

T=49°C线性回归乘幂回归U二aUb21指数回归U二bexp(eU]/kT)U=2aU+b1U/V1U/V2U*/V2(U-U*)222/V2U*/V2(U-U*)222/V2U*/V2(U-U*)2/V22210.250.108-1.3402.0970.0951.69E-040.1091.00E-0620.260.158-0.7630.8490.1471.21E-040.1564.00E-0630.270.228-0.1860.1720.2243.14E-030.2241.60E-0540.280.3180.3900.0050.3363.24E-040.3232.50E-0550.290.460.9670.2570.4971.37E-030.4641.60E-0560.30.6631.5440.7760.7243.72E-030.6671.60E-0570.310.9682.1211.3291.0435.63E-030.9598.10E-0580.321.3782.6981.7411.4851.14E-021.3791.00E-0690.331.9873.2741.6572.0911.08E-021.9831.60E-05100.342.8993.8510.9072.9142.25E-042.8512.30E-03110.354.1674.4280.0684.0242.04E-024.0994.62E-03120.365.9245.0050.9195.5051.76E-015.8939.61E-04130.378.2985.58167.3797.4