Units cancelling

A note on good practice Note that although, as usual, we have left the numerical calculation to a single, final step, the same is not true for the units canceling units docs not introduce rounding errors and clarifies each step. 

First of all, the units cancel to give moles and atoms, which are the units that the problem asked for. Further, the mass of Hg is quite small, so we expect the number of moles to be small also. The number of atoms, 1.08 X lO , is large but much smaller than Avogadro s number. 

The calculation of carbon is shown in detail to highlight the unit cancellation. Here are the... 

You should verily that the units cancel properly, giving a result in pressure units. The problem describes a large amount of methane in a relatively small volume, so a high pressure is reasonable. This high value indicates why gases such as methane must be stored in tanks made of materials such as steel that can withstand high pressures. 

The first four steps of the seven-step strategy are identical to the ones in Example. In this example, addition of a strong acid or base modifies the concentrations that go into the buffer equation. We need to determine the new concentrations (Step 5) and then apply the buffer equation (Step 6). In dealing with changes in amounts of acid and base, it is often convenient to work with moles rather than molarities. The units cancel in the concentration term of the buffer equation, so the ratio of concentrations can be... 

Since the mole fraction is a ratio of moles (of one substance) to moles (total), the units cancel out and the mole fraction has no units. 

Ans. Mole fraction has no units. It is defined as one number of moles divided by another, and the units cancel. 

We can omit the units of the two pressures on the left-hand side because Equation (5.5) is written as a ratio, so the units cancel we require only a relative change in pressure. 

Note how the units cancel to yield a dimensionless mole fraction. 

In working problems, be sure that your units cancel. 

The new pressure is greater than the original pressure, making the answer a reasonable one. Note that all the units canceled except atm, which is the unit that you wanted. 

Note that the units canceled, leaving the desired volume unit of liters. Overall, the volume did increase, so in this case the pressure decrease had a greater effect than the temperature decrease. This seems reasonable, looking at the numbers. There is a relatively small change in the Kelvin temperature (293 K versus 263 K) compared to a much larger change in the pressure (760.0 torr versus 450.0 torr). 

Tips—Make sure the temperature is in kelvin gas laws are being applied to gases only the units cancel and the answer is reasonable. 

Be sure your units cancel giving you the unit desired in the final answer. 

Make sure your units cancel, leaving you with the units desired in your final answer. Round off your final numerical answers to the correct number of significant figures. Remember, most molecular compounds—compounds containing only nonmetals—do not ionize in solution. Acids are the most common exceptions. 

When working mathematical problems, be sure your units cancel to give you the desired unit in your answer. 

Make sure your units cancel in your calculations, leaving the unit you want. 

Now, solve the problem using the dimensional analysis method. We want the answer to be in inches per second. Set up the fractions with inches on the top and seconds on the bottom, so that the centimeter and minute units cancel. 

The second units cancel and the inch units cancel as shown above. 

Multiply the given quantity by conversion factors until all the unwanted units cancel out and only the desired units remain. 

As you become more familiar with this type of question, you will be able to complete more than one step at once. Below, you can see how the conversion factors we used in each step above can be combined. Set these conversion ratios so that the units cancel out correctly. 

The units in this problem are all over the place. The question involves seconds the rate is in miles and hours and one of the dimensions is in feet. First, change everything to seconds and feet. Then use the formula for the height. To change the miles per hour to feet per second, multiply miles per hour x hour per seconds x feet per mile. Notice how the units cancel to give you the rate you want. 

Dimensional analysis is based on multiplication of improper fractions. The key to using it is to keep the units with the numbers. The units cancel out in much the same way as fractions do when multiplying. Writing out your work using dimensional analysis makes it much easier to keep track of what you are doing, and to double check your work. If your units do not properly cancel out to give you the unit that you want for the answer, your answer is wrong. 

You then need to write the other conversions as fractions so that the units cancel out ... 

Remember, any whole number that you use as a conversion can be written as a fraction either over 1 or 1 over itself You need to make sure the units cancel out. Once you cancel the units you do the math in the same way that you do multiplication of fractions. You multiply the top numbers together, in this case 45 x 5,280 x 1 x 1 = 237,600, and the bottom numbers together, 1 x 1 x 60 x 60 = 3,600, and divide the top number by the bottom number. The result can be written over 1, or can be written in a more common form as 66 feet / second (fjps). 

Adsorption onto Particles. The Gibbs Adsorption law relates how adsorption onto surfaces affects interfacial tension, dy = - RTfd In c. where y = intcrfacial or surface tension, in N/m (I N/m = 1000 dyn/cm) R = gas constant T = absolute temperature T = interfacial or surface concentration, m mol/unil area (i.c.. adsorption) and i = dimensionless concentration (d In r- = t/r/r, thus units cancel). 

The dimensional-analysis method gives the right answer only if the equation is set up so that the unwanted units cancel. If the equation is set up in any other way, the units won t cancel properly, and you won t get the right answer. Thus, if you were to multiply your height in inches by the incorrect conversion factor inches per meter, you would end up with an incorrect answer expressed in meaningless units ... 

The known information is the speed in kilometers per hour the unknown is the speed in miles per hour. Find the appropriate conversion factor, and use the dimensional-analysis method to set up an equation so the km units cancel. 

When you become more confident in working multiple conversion problems, you can set up one large equation in which all unwanted units cancel ... 

Because many experiments involve numerical calculations, it s often necessary to manipulate and convert different units of measure. The simplest way to carry out such conversions is to use the dimensional-analysis method, in which an equation is set up so that unwanted units cancel and only the desired units remain. It s also important when measuring physical quantities or carrying out calculations to indicate the precision of the measurement by rounding off the result to the correct number of significant figures. 

Begin with the known information, and set up an equation using appropriate conversion factors so that the unwanted units cancel. In the present instance, let s begin with the width of the pencil line in millimeters, then convert to meters, and then divide the line width in meters by the diameter of a single atom in meters. 

The problem gives the mass of sucrose and asks for a mass-to-mole conversion. Use the molar mass of sucrose as a conversion factor, and set up an equation so that the unwanted unit cancels. 

Values of Kc are generally reported without units because the concentrations in the equilibrium constant expression are considered to be concentration ratios in which the molarity of each substance is divided by its molarity (1 M) in the thermodynamic standard state (Section 8.6). Because the units cancel, the concentration ratios and the values of Kc are dimensionless. For experiment 1 in Table 13.1, for example,... 

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