Units and Conversion Factors in Calculations

All measured quantities consist of a number and a unit a person s height is 6 feet, not 6. Ratios of quantities have ratios of units, such as miles/hour. (We discuss the most important units in chemistry in the next section.) To minimize errors, try to make a habit of including units in all calculations. 

The arithmetic operations used with measured quantities are the same as those used with pure numbers in other words, units can be multiplied, divided, and canceled  

Conversion factors are ratios used to express a measured quantity in different units. Suppose we want to know the distance of that 150-mile car trip in feet. To convert the distance between miles and feet, we use equivalent quantities to construct the desired conversion factor. The equivalent quantities in this case are 1 mile and the number of feet in 1 mile  

We can construct two conversion factors from this equivalency. Dividing both sides by 5280 ft gives one conversion factor (shown in blue)  

It s very important to see that, since the numerator and denominator of a conversion factor are equal, multiplying by a conversion factor is the same as multiplying by 1. Therefore, even though the number and unit of the quant it change, the size of the quantit remains the same. 

dividing both sides by 1 mi gives the other conversion factor (the inverse)  

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Units and Conversion Factors in Calculations 10 A Systematic Approach to Solving Chemistry Problems 12... 

The authoritative values for physical constants and conversion factors used in thermodynamic calculations are assembled in Table 2.3. Furthermore, information about the proper use of physical quantities, units, and symbols can be found in several additional sources [5]. 

When C2D2 frequencies are used, in should be replaced by /Hq. The force constants for acetylene can be calculated from these relations using the measured vibrational frequencies, and the bond lengths can be determined from the rotational analysis described below. If one expresses the frequencies in cm units and the masses in appropriate isotopic mass units, the factors 4t7 should be replaced by 4t7 c 10 NY = 5.8918 X 10 (this includes a factor of 10 kg/g mass conversion). This substitution gives the force constants Y, kg, and Yr in N m units and the bending constants kg and kgg in units of N m. 

Using the result obtained by Bull and Gortner [J. Phys, Chem, 36, 111 (1932)], that SIP for the streaming of 2 X 10 N sodium chloride through a diaphragm of quartz particles is about 25 millivolts per cm. of mercury pressure, calculate the approximate specific surface conductance of the solution used. Compare the result with the normal value for sodium chloride at the same concentration. The viscosity and dielectric constant of the solution may be assumed to be the same as for water, and the zeta-potential may be taken as 0.05 volt. (Care should be exercised in the matter of units, use being made of the conversion factors in Table I.)... 

Example 1.1 demonstrates how conversion factors are calculated from the exact numbers used to set up the definitions of units in the SI and fps systems. In conversions involving g in fps units, the use of the exact numerical ratio 9.80665/0.3048 in place of the fps number 32.1740 is recommended to give... 

Such strange units should signal that you made an error in setting up the conversion factor. In addition, the answer, 6.67, is less than 10.0, whereas the balanced equation shows that more moles of O2 than of CU2S are needed. Be sure to think through the calculation when setting up the conversion factor and canceling units. 

It is now possible to calculate unit costs of conversion for comparison with sales price per unit. For a single product, project unit costs are simply calculated by dividing total annual production costs by the number of" units produced. Unit costs therefore reflect capacity utilization, and this is a critical factor in calculating unit costs. For multiple-product projects without any historical data on which to base fixed cost allocations, it is generally better to only calculate the contribution per unit (sales revenue less variable cost) to covering fixed costs and profit. 

Table 1.9 gives some unit conversion factors calculated from the information given in Tables 1.6-1.8 and in preceding parts of this chapter. Note that, in each case, two unit conversion factors are calculated the one that is used depends upon the units that are required for the answer. 

The dose conversion factor, DCF, in effective dose per exposure unit is calculated by taking into account the dose function of the particle diameter (Figure 5.4) and the radon decay product characteristics. The dose conversion factors for living and work places with typical activity size distributions, as a function of the unattached fraction, fp, are illustrated in Figures 5.13 and 5.14, respectively. The value of the dose conversion factor, DCFae, for fp = 0 represents the dose contribution of the radon product aerosols. The values of dose conversion factor in Figure 5.14 are based on aerosol conditions which are typical for many workplaces. 

Conversion factors are ratios of equivalent quantities having different units they are used in calculations to change the units of quantities. Decimal prefixes and exponential notation are used to express very large or very small quantities. (Section 1.3)... 

An observed spectroscopic transition in the hydrogen atom involves the 2/ Is transition. Using Eq. (4-8), evaluate this energy difference in units of hertz (Hz). (1 Hz= 1 s ) Do the calculation using both me and n. (See Appendix 10 for constants and conversion factors.) How much error in this calculation, in parts per million, is introduced by ignoring the finite mass of the nucleus (i.e., using We instead of /u-) ... 

Unear Umts. The following procedure is used for converting linear units to the proper number of significant places the maximum and minimum limits in inches are calculated. The corresponding two values are converted exacdy into millimeters by multiplying each by the conversion factor 1 in. = 25.4 mm. The results are rounded in accordance with Table 4. 

The water-vapor transmission rate (WVTR) is another descriptor of barrier polymers. Strictly, it is not a permeabihty coefficient. The dimensions are quantity times thickness in the numerator and area times a time interval in the denominator. These dimensions do not have a pressure dimension in the denominator as does the permeabihty. Common commercial units for WVTR are (gmil)/(100 in. d). Table 2 contains conversion factors for several common units for WVTR. This text uses the preferred nmol/(m-s). The WVTR describes the rate that water molecules move through a film when one side has a humid environment and the other side is dry. The WVTR is a strong function of temperature because both the water content of the air and the permeabihty are direcdy related to temperature. Eor the WVTR to be useful, the water-vapor pressure difference for the value must be reported. Both these facts are recognized by specifying the relative humidity and temperature for the WVTR value. This enables the user to calculate the water-vapor pressure difference. Eor example, the common conditions are 90% relative humidity (rh) at 37.8°C, which means the pressure difference is 5.89 kPa (44 mm Hg). 

The Excel spreadsheet is constructed so that on page one, the referenced properties are listed in Column C, and the same with conversion factors to SI units in Column D. Conversion formulas and values calculated in SI Units are in Column E. Column F is a duplicate of Column E, and this can be used for additional calculation by changing to other conditions or to an entirely new case. It is recommended toleave Column E alone for a comparison case and to copy Column F to another page to execute calculations. 

Toxicity alucs for carcinogenic effects also can be c.xprcsscd in terms of risk per unit concentration of the substance in the medium where human contact occurs. These measures, called unit risks, are calculated by dividing the slope factor by 70 kg and multiplying by the inhalation rate (20 m /day) or the water consumption rate (2 L/day), respecti ely, for risk associated with unit concentration in air or water. Where an absorption fraction less than 1.0 has been applied in deriving the slope factor, an additional conversion factor is necessary in the calculation of unit risk so that the unit risk will be on an administered dose basis. The standardized duration assumption for unit risks is understood to be continuous lifetime c.xposure. Hence, when there is no absorption conversion required ... 

Table 0.1 shows such atomic units . The accepted values of the SI constants are themselves subject to minor experimental improvements, so authors generally report (he results of molecular modelling calculations as (e.g.) R = 50aa and give the conversion factor to SI somewhere in their paper, usually as a footnote. 

This equation can be used to calculate the energy change in joules, if you know Am in kilograms. Ordinarily, Am is expressed in grams AE is calculated in kilojoules. The relationship between AE and Am in these units can be found by using conversion factors ... 

The more permeable component is called the fast gas, so it is the one enriched in the permeate stream. Permeabihly through polymers is the product of solubility and diffusivity. The diffusivity of a gas in a membrane is inversely proportional to its kinetic diameter, a value determined from zeolite cage exclusion data (see Table 20-26 after Breck, Zeolite Molecular Sieves, Wiley New York, 1974, p. 636). Tables 20-27, 20-28, and 20-29 provide units conversion factors useful for calculations related to gas-separation membrane systems. 

A record of all calculations performed in connection with the test, including units of measure, conversion factors, and equivalency factors. 

No deaths or evidence of toxicity were attributable to diisopropyl methylphosphonate administered for 26 weeks in the drinking water of rats at concentrations of 0.6 ppb, 6.0 ppb, 10 ppm, and 1,000 ppm (6.6x 10"7, 6.6x 10"5, 0.011, and 1.1 mg/kg/day, respectively) (Army 1978). It should be noted that there is some confusion concerning the concentration units used in this study (EPA 1989). EPA (1989) states that conversions between ppm and mg/L were incorrectly calculated using the air conversion factor. 

It is usually the best practice to work through design calculations in the units in which the result is to be presented but, if working in SI units is preferred, data can be converted to SI units, the calculation made, and the result converted to whatever units are required. Conversion factors to the SI system from most of the scientific and engineering units used in chemical engineering design are given in Appendix D. 

It should be noted that a dimensional analysis of this problem results in one more dimensionless group than for the Newtonian fluid, because there is one more fluid rheological property (e.g., m and n for the power law fluid, versus fi for the Newtonian fluid). However, the parameter n is itself dimensionless and thus constitutes the additional dimensionless group, even though it is integrated into the Reynolds number as it has been defined. Note also that because n is an empirical parameter and can take on any value, the units in expressions for power law fluids can be complex. Thus, the calculations are simplified if a scientific system of dimensional units is used (e.g., SI or cgs), which avoids the necessity of introducing the conversion factor gc. In fact, the evaluation of most dimensionless groups is usually simplified by the use of such units. 

Then the unattached fraction was calculated in each measurement and was found to be between. 05 and. 15 without aerosol sources in the room and below. 05 in the presence of aerosol sources. The effective dose equivalent was computed with the Jacobi-Eisfeld model and with the James-Birchall model and was more related to the radon concentration than to the equilibrium equivalent radon concentration. On the basis of our analysis a constant conversion factor per unit radon concentration of 5.6 (nSv/h)/(Bq/m ) or 50 (ySv/y)/(Bq/m3) was estimated. 

The next two steps in the procedure of Leonard and Ashman are the conversion of the diagonal elements from atomic units into force field units and calculation of scaling factors for bond lengths and angles. The calculated force constants had to be scaled down by approximately 25% and 70% to yield force constants comparable in numerical size with those included in MM2. Neither force constants nor scaling factors can be incorporated directly into a different force field. A modification of the described procedure that meets the requirements of CVFF was developed. Fragments with known force field parameters were chosen. After a full geometry optimization (HF/6-31G ) second derivatives and vibrational frequencies were calculated. The force... 

Still other units encountered in the literature and workplace come from various other systems (absolute and otherwise). These include metric systems (c.g.s. and MKS), some of whose units overlap with SI units, and those (FPS) based on English units. The Fahrenheit and Rankine temperature scales correspond to the Celsius and Kelvin, respectively. We do not use these other units, but some conversion factors are given in Appendix A. Regardless of the units specified initially, our approach is to convert the input to SI units where necessary, to do the calculations in SI units, and to convert the output to whatever units are desired. 

For quantitative considerations it is convenient to use atomic units (a.u.), in which h = eo = me = 1 (me is the electronic mass) by definition. They are based on the electrostatic system of units so Coulomb s law for the potential of a point charge is = q/r. Conversion factors to SI units are given in Appendix B here we note that 1 a.u. of length is 0.529 A, and 1 a.u. of energy, also called a hartree, is 27.211 eV. Practically all publications on jellium use atomic units, since they avoid cluttering equations with constants, and simplify calculations. This more than compensates for the labor of changing back and forth between two systems of units. 

Dimensional analysis, sometimes called the factor label (unit conversion) method, is a method for setting up mathematical problems. Mathematical operations are conducted with the units associated with the numbers, and these units are cancelled until only the unit of the desired answer is left. This results in a setup for the problem. Then the mathematical operations can efficiently be conducted and the final answer calculated and rounded off to the correct number of significant figures. For example, to determine the number of centimeters in 2.3 miles ... 

This section presents sample calculations to aid the reader in understanding the calculations behind the development of a fuel cell power system. The sample calculations are arranged topically with unit operations in Section 10.1, system issues in Section 10.2, supporting calculations in Section 10.3, and cost calculations in Section 10.4. A list of conversion factors common to fuel cell systems analysis is presented in Section 10.5 ans a sample automotive design calculation is presented in Section 10.6. 

For chemical reactions and phase transformations, the energy absorbed or liberated is measured as heat. The principal unit for reporting heat is the calorie, which is defined as the energy needed to raise the temperature of 1 gram of water at l4.5° C by a single degree. The term kilocalorie refers to 1,000 calories. Another unit of energy is the joule (rhymes with school), which is equal to 0.239 calories. Conversely, a calorie is 4.184 joules. The translation of calories to joules, or kilocalories to kilojoules, is so common in chemical calculations that you should memorize the conversion factors. 

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