Logarithms rounding

A pH number is a logarithm. Table 17.4 shows the logarithms of 3.45 multiplied by five different powers of 10 0, 1, 2, 8, and 12. One column shows the value of the logarithm to seven decimals. Another shows the logarithms rounded off to the correct number of significant figures. Notice three things ... 

A note on good practice Exponential functions (inverse logarithms, e ) are very sensitive to the value of x, so carry out all the arithmetic in one step to avoid rounding errors. 

Note To find e-95 66, take the inverse In of-95.66 on your calculator, inv In of-95.66 = 2.85 x 10 42. Keep one more significant figure and round off to three significant figures at the end, particularly when working with logarithms. 

Note Keep all the significant figures and round at the end. Remember the number of decimal places in pH or pOH values are set by the number of significant figures in the [H+] or [OH-] this is a result of working with logarithms. 

A Calculations containing logarithms and/or differences in reciprocals are very sensitive to rounding. Be careful not round intermediate values. 

Rate Constant. The logarithms of second-order rate constants (expressed as M sec.-1) are given for 25°C. unless otherwise indicated. These are rounded off to the nearest 0.1 unit. The ionic strengths are often 0.1-0.2 M. 

To reach steady state, the residence time of the fluid in a constant stretch rate needs to be sufficiently long. For some polymer melts, this has been attained however, for polymer solutions this has proved to be a real challenge. It was not until the results of a world wide round robin test using the same polymer solution, code named Ml, became available that the difficulties in attaining steady state in most extensional rheometers became clearer. The fluid Ml consisted of a 0.244% polyisobutylene in a mixed solvent consisting of 7% kerosene in polybutene. The viscosity varied over a couple of decades on a logarithmic scale depending on the instrument used. The data analysis showed the cause to be different residence times in the extensional flow field... 

In this unit you will find explanations, examples, and practice dealing with the calculations encountered in the chemistry discussed in this book. The types of calculations included here involve conversion factors, metric use, algebraic manipulations, scientific notation, and significant figures. This unit can be used by itself or be incorporated for assistance with individual units. Unless otherwise noted, all answers are rounded to the hundredth place. The calculator used here is a Casio FX-260. Any calculator that has a log (logarithm) key and an exp (exponent) key is sufficient for these chemical calculations. 

Here we give the numerical results for the energy levels in table 1 for the different orders in a. At the end of each equation we have rounded the numerical results. As uncertainties for the S -levels we took 50% of the leading logarithmic contribution in the order R a5, for the P-levels we took 10% of the known order R a4, for the uncertainty of the 3 )-levels (where the order Rc a4 is yet... 

The scaling of round-trip times is shown in Fig. 3 for the two-dimensional fully frustrated Ising model. The power-law slowdown of round-trip times for the flat-histogram ensemble is reduced to a logarithmic correction... 

Fig. 3. Scaling of round-trip times for a random walk in energy space sampling a flat histogram open squares) and the optimized histogram solid circles) for the two-dimensional fully frustrated Ising model. While for the multicanonical simulation a power-law slowdown of the round-trip times 0 N L ) is observed, the round-trip times for the optimized ensemble scale like 0([A ln A ] ) thereby approaching the ideal 0(A )-scaling of an unbiased Markovian random walk up to a logarithmic correction...

Be especially careful in rounding the results of calculations involving logarithms. The following rules apply to most situations. These rules are illustrated in Example 6-8. 

For a purely monolayer model, the exact Onsager [82] solution for the two-dimensional lattice problem, predicts the so-called logarithmic discontinuity on the heat capacity curve at the critical temperature T. The deviations of the actual adsorption systems from the ideal Onsager model mean that, instead of a logarithmic discontinuity, more or less rounded peaks are observed, centered at T = Tc. The surface energetic heterogeneity is believed to be the main source of that rounding. 

Nephrolithiasis occurs in 10% to 25% of patients with gout. Factors that predispose individuals to uric acid nephrolithiasis include excessive urinary excretion of uric acid, an acidic urine, and a highly concentrated urine. The risk of renal calcuh approaches 50% in individuals whose renal excretion of uric acid exceeds 1100 mg/day. In addition to pure uric acid stones, hyperuricosuric individuals are at increased risk for mixed uric acid-calcium oxalate stones and pure calcium oxalate stones. Uric acid stones are usually small, round, and radiolucent. Uric acid stones containing calcium are radiopaque. Uric acid has a negative logarithm of the acid ioiuzation constant of 5.5. Therefore when the urine is acidic, uric acid exists primarily in the un-ionized, less soluble form. At a urine pH of 5.0, urine is saturated at a uric acid level of 15 mg/dL. When the urine pH is 7.0, the solubility of uric acid in urine is increased to 200 mg/dL. In patients with uric acid nephrolithiasis, urinary pH typically is less than 6.0 and frequently less than 5.5. When an acidic urine is saturated with uric acid, spontaneous precipitation of stones may occur. 

Permeation Properties. The data shown in Figure 2 are the toluene permeation rates of the fluorinated and untreated containers g. toluene/container per day are plotted vs. the time of toluene exposure on a logarithmic scale. These cumulative permeation rates were calculated based on the cumulative weight loss over the time of toluene exposure, as opposed to the differential permeation rates based on the differential weight loss over each time interval. The room temperature permeation rates for the in-situ fluorinated containers were less than 0.01 g./day and, hence, have been rounded up to 0.01 g./day for illustrative purposes. In Figure 2, the,flat portion of the curves for the untreated containers yielded the steady state permeation rates. From these values, the permeability coefficients (P) for the untreated containers were calculated using Equation 1. 

It is frequently convenient to take measurements at nominally equal logarithmic time intervals, and even more convenient if this can be done automatically. A suggested framework for this is given in the standards as in the table. In practice, how cver. where manual logging of data is undertaken, it is generally more convenient to w ork in whole numbers of days and weeks than in arbitrarily round numbers of hours. 

Figure 8.6 Double logarithmic plot of delamination rate 6a/dN versus Gin,ax and Gnnuix. respectively, from cyclic fatigue (fat) with an R-ratio of 0.1 of carbon fibre epoxy (IM7/977-2) under mode I and mode II from ESIS TC4 round robins (2009, 2012), and under fixed-ratio mixed mode I/II from preliminary single-specimen testing (black dots) at the author s laboratory expected values for fixed-ratio mixed mode VYL (mixed mode I/n fat expect) were calculated from cyclic fatigue mode I and mode H round-robin data (2009 and 2012, respectively), quasistatic (static) and cyclic fatigue mode I and mode H values from literature [39] (nsr = no shear reversal sr = with shear reversal), and earlier cyclic fatigue mode H [60] are shown for comparison.

Figure 8.7 Exponent m of the power-law fit (A Gm) to the double logarithmic plot of delamination rate daldtN versus Gi ax from cyclic mode I fatigue with an R-ratio of 0.1 of five specimens of carbon fibre epoxy (IM7/977-3) from the ASTM round robin.

Figure 8.10 Double logarithmic plot of delamination rate da/AN versus Gi ax and Gnn,ax> respectively, from cyclic fatigue with an R-ratio of 0.1 of a carbon fibre thermoplastic (AS4/ PEEK) under mode I loading from an ESIS TC4 round robin. The min and max lines indicate the scatter band of tests from five laboratories cyclic fatigue mode I and mode II values [59] and averaged quasi-static mode I and mode II values [52] are shown for comparison (s.d. = standard deviation).

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