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221Midterm.Equations

# 221Midterm.Equations - Constants R = 8.314 J-K-1-mol-1 N0 =...

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Constants R = 8 . 314 J-K - 1 -mol - 1 k B = 1 . 381 × 10 - 23 J-K - 1 e = 1 . 1602 × 10 - 19 C m e = 9 . 11 × 10 - 31 kg N 0 = 6 . 022 × 10 23 mol - 1 h = 6 . 626 × 10 - 34 J-s c = 2 . 998 × 10 8 m-s - 1 0 = 8 . 854 × 10 - 12 C 2 N - 1 m - 2 Related Information Quadratic Equation: if ax 2 + bx + c = 0 , then x = - b ± b 2 - 4 ac 2 a Standard Deviation: s = ( x i - x ) 2 N - 1 Error Propagation Equation: σ 2 y ( u, v, . . . ) = σ 2 u dy du 2 + σ 2 v dy dv 2 + · · · t-tests: True Mean Comparison: x 1 - x 2 = ± ts/ N Measured Means Comparison (when σ 2 1 = σ 2 2 ): x 1 - x 2 = ± ts pooled N 1 + N 2 N 1 N 2 s pooled = s 2 1 ( N 1 - 1) + s 2 2 ( N 2 - 1) N 1 + N 2 - 2 Measured Means Comparison (when σ 2 1 = σ 2 2 ): | x 1 - x 2 | = ± t s 2 1 /N 1 + s 2 2 /N 2 ν = ( s 2 1 /N 1 + s 2 2 /N 2 ) 2 ( s 2 1 /N 1 ) 2 N 1 + 1 + ( s 2 2 /N 2 ) 2 N 2 + 1 - 2 Degrees of Freedom, ν t(90%) t(95%) t(99%) 1 6.314 12.706 63.657 2 2.920 4.303 9.925 3 2.353 3.182 5.841 4 2.132 2.776 4.604 5 2.015 2.571 4.032 6 1.943 2.447 3.707 7 1.895 2.365 3.500 8 1.860 2.306 3.355 9 1.833 2.262 3.250 10 1.812 2.228 3.169 15 1.7531 2.131 2.947 20 1.725 2.086 2.845 25 1.708 2.060 2.787 30 1.697 2.042 2.750 40 1.684 2.021 2.704 60 1.671 2.000 2.660 1.64 1.96 2.576 Table 1: Values of t for ν degrees of freedom for 95% confidence levels. ν = N - 1 = degrees of freedom, and confidence limit = x ± ts/ N . 2

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ν 1 = 1 2 3 4 5 6 7 8 9 10 12 15 20 30 40 60 ν 2 = 1 161.448 199.500 215.707 224.583 230.162 233.986 236.768 238.883 240.543 241.882 243.906 245.950 248.013 250.095 251.143 252.196 254.314 2 18.5128 19.0000 19.1643 19.2468 19.2964 19.3295 19.3532 19.3710 19.3848 19.3959 19.4125 19.4291 19.4458 19.4624 19.4707 19.4791 19.4957 3 10.1280 9.5521 9.2766 9.1172 9.0135 8.9406 8.8867 8.8452 8.8123 8.7855 8.7446 8.7029 8.6602 8.6385 8.5944 8.5720 8.5264 4 7.7086 6.9443 6.5914 6.3882 6.2561 6.1631 6.0942 6.0410 5.9988 5.9644 5.9117 5.8578 5.8025 5.7744 5.7170 5.6877 5.6281 5 6.6079 5.7861 5.4095 5.1922 5.0503 4.9503 4.8759 4.8183 4.7725 4.7351 4.6777 4.6188 4.5581 4.5272 4.4638 4.4314 4.3650 6 5.9874 5.1433 4.7571 4.5337 4.3874 4.2839 4.2067 4.1468 4.0990 4.0600 3.9999 3.9381 3.8742 3.8415 3.7743 3.7398 3.6689 7 5.5914 4.7374 4.3468 4.1203 3.9715 3.8660 3.7870 3.7257 3.6767 3.6365 3.5747 3.5107 3.4445 3.4105 3.3404 3.3043 3.2298 8 5.3177 4.4590 4.0662 3.8379 3.6875 3.5806 3.5005 3.4381 3.3881 3.3472 3.2839 3.2184 3.1503 3.1152 3.0428 3.0053 2.9276 9 5.1174 4.2565 3.8625 3.6331 3.4817 3.3738 3.2927 3.2296 3.1789 3.1373 3.0729 3.0061 2.9365 2.9005 2.8259 2.7872 2.7067 10 4.9646 4.1028 3.7083 3.4780 3.3258 3.2172 3.1355 3.0717 3.0204 2.9782 2.9130 2.8450 2.7740 2.7372 2.6609 2.6211 2.5379 11 4.8443 3.9823 3.5874 3.3567 3.2039 3.0946 3.0123 2.9480 2.8962 2.8536 2.7876 2.7186 2.6464 2.6090 2.5309 2.4901 2.4045 12 4.7472 3.8853 3.4903 3.2592 3.1059 2.9961 2.9134 2.8486 2.7964 2.7534 2.6866 2.6169 2.5436 2.5055 2.4259 2.3842 2.2962 13 4.6672 3.8056 3.4105 3.1791 3.0254 2.9153 2.8321 2.7669 2.7144 2.6710 2.6037 2.5331 2.4589 2.4202 2.3392 2.2966 2.2064 14 4.6001 3.7389 3.3439 3.1122 2.9582 2.8477 2.7642 2.6987 2.6458 2.6022 2.5342 2.4630 2.3879 2.3487 2.2664 2.2229 2.1307 15 4.5431 3.6823 3.2874 3.0556 2.9013 2.7905 2.7066 2.6408 2.5876 2.5437 2.4753 2.4034 2.3275 2.2878 2.2043 2.1601 2.0658 16 4.4940 3.6337 3.2389 3.0069 2.8524 2.7413 2.6572 2.5911 2.5377 2.4935 2.4247 2.3522 2.2756 2.2354 2.1507 2.1058 2.0096 17 4.4513 3.5915 3.1968 2.9647 2.8100 2.6987 2.6143 2.5480 2.4943 2.4499 2.3807 2.3077 2.2304 2.1898 2.1040 2.0584 1.9604 18 4.4139 3.5546 3.1599 2.9277 2.7729 2.6613 2.5767 2.5102 2.4563 2.4117 2.3421 2.2686 2.1906 2.1497 2.0629 2.0166 1.9168 19 4.3807 3.5219 3.1274 2.8951 2.7401 2.6283 2.5435 2.4768 2.4227 2.3779 2.3080 2.2341 2.1555 2.1141 2.0264 1.9795 1.8780 20 4.3512 3.4928
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221Midterm.Equations - Constants R = 8.314 J-K-1-mol-1 N0 =...

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