Cancellation of units

Due to a fortuitous cancellation of units, volume can be expressed in cm3 and pressure in MPa to give the same results as volume in m3 with pressure in Pa, without invoking any conversion factors. Thus pV in Pa m3 gives Joules as does pV in MPa cm3. 

To convert from one unit to another, we multiply by the ratio that leads to an appropriate cancellation of units. For example, a volume of exactly two liters is expressed in quarts as follows ... 

To summarize, the amounts of different reagents that participate in a chemical reaction are related through the stoichiometric coefficients in the balanced chemical equation. To convert from moles of one reagent to moles of any other reagent, multiply by the stoichiometric ratio that leads to proper cancellation of units ... 

In the interest of clarity, cancellation of units will be omitted in the... 

The functions of the above conversion factors are convert starting information from grams to moles using the coefficients in the balanced equation convert to moles of desired substance convert from moles to grams of desired substance. Be sure to label each factor completely with its units. Then if canceling of units does not result in the correct units for the answer you will know that there has been an error in your reasoning. 

Using the density as a conversion factor from volume to mass, we have m = (10.0jnfc+Ig)(l3.6- j Cancellation of units results in ... 

Note once again that dimensional analysis (cancellation of units) results in the desired units of mmol/mL. 

From here on, except in particularly complex situations, the canceling of units in calculations is no longer shown. 

The cancellation of units provides the basis for choosing which conversion factor is needed It is always the one that allows the unit being converted to be canceled and leaves the new unit uncanceled. 

Many kinds of numerical problems in everyday life as well as in science can be solved by extending the use of conversion factors and cancellation of units beyond unit conversion. The use of density expressed as mass per unit volume is a simple example of such an extension. Density provides the connection between mass and volume. Given that the density of lead is 11.4 g/cm , you can find the mass in grams of a piece of lead of known volume or the volume of a piece of lead of known mass. If the known information is that a piece of lead has a volume of 25.0 cm and the unknown information is its mass, the problem is set up and solved as follows... 

Then we can calculate the moles of ethanol by proper cancellation of units, taking into account the density of ethanol and its molar mass (46.0 g/mol) ... 

This potential problem is less likely if you enter the data as milligrams and millimoles. If you recognize that 0.1 g of alkene is 100 mg and 0.5 mL of HBr solution contains 358 mg of HBr (0.5 mL X 1490 mg/mL X 0.48), the entries would be those shown in Table 1.2. You may then determine the limiting reagent and calculate the theoretical yield as in Example 1. Note that the necessary cancellation of units occurs when the molar mass is expressed in mg/mmol. 

The conversion factors give correct units (inches). An inch is much shorter than a mile, so it makes sense that the number of inches is much larger than the number of miles. Cancellation of units will be omitted In the remainder of this book, but you may find it useful to continue cancelling units. 

Cancellation of units leads to the correct final units (mL). A density less than one (0.789 g/mL) tells us that one mL has a mass less than one gram. Thus the number of mL needed must be numerically greater than the number of grams required. 

Think About It Again, check the cancellation of units carefully and make sure that the magnitudes of your results are reasonable. 

Think About It Look carefully at the cancellation of units and make sure that the number of kilojoules is smaller than the number of joules. It is a common error to multiply by 1000 instead of dividing in conversions of this kind. 

When you use/ in a calculation, use the version that Vitales proper cancellation of units. 

If you check the cancellation of units carefully in ideal gas equation problems, however, then with practice you will develop a sense of whether such a calculated volume is reasonable. 

At the heart of the dimensional analysis method is a technique known as the cancellation of units. The basic rule is when the same unit appears in both the numerator and denominator, these units cancel. This cancellation of units is analogous to the cancellation of numbers when the same number appears in both the numerator and denominator. 

Think About It Careful cancellation of units is essential. Note that this amount of CO2 corresponds to 162 moles or 3.9 kg of LiOH. (IPs a good idea to verify this yourself.)... 

The first term in the denominator, after simple cancellation of units becomes... 

If, in Example 1-4, we had not been careful about the cancellation of units and had multiplied by percentage (3.5/100) instead of dividing by it (100/3.5), we would have obtained the numerical answer 2.5 X 10. This would be a 2.5 mL sample of seawater, weighing about 2.5 g. Clearly, a sample of seawater that contains 75 g of sodium chloride must have a mass greater than 75 g. 

In calculations requiring the Avogadro constant, students often ask when to multiply and when to divide by. One answer is always to use the constant in a way that gives the proper cancellation of units. Another answer is to think in terms of the expected result. In calculating a number of atoms, we expect the answer to be a very large number and certainly never smaller than one. The number of moles of atoms, conversely, is generally a number of more modest size and will often be less than one. 

---