Problem 19 - Introductory Chemistry Atoms First

Question

$$ 27.5 \text { in. } / 2.0 \mathrm{~h}=? $$

Step-by-Step Solution

Answer

The rate is \(13.75~\frac{\text{in.}}{\text{h}}\).

Step 1: Identify the given values

We have the values 27.5 inches and 2.0 hours.

Step 2: Set up the division

Now, set up the division as follows: \(\frac{27.5 \text{ in.}}{2.0 \text{ h}}\).

Step 3: Perform the division

Divide 27.5 by 2.0: \[\frac{27.5}{2.0} = 13.75\]

Step 4: Write the final answer

The result of the division is 13.75. So, the answer is: \(13.75~\frac{\text{in.}}{\text{h}}\). The rate is 13.75 inches per hour.

Key Concepts

Unit Conversion

Understanding unit conversion is essential when dealing with measurements in various physical quantities. Essentially, it is the process of converting the value of a unit to another within the same measurement system or to a different system. Converting units becomes critical when you need to compare, add, or subtract quantities with different units.

For example, in the exercise \(27.5 \text{ in. } / 2.0 \mathrm{~h}= ?\), to make sense of the rate, you might need to convert inches to feet or meters if required by the context in which you are working. To convert units, you can use conversion factors, which are ratios that express how many of one unit are equivalent to another. The key to successful unit conversion is making sure that the units you don't want cancel out, leaving only the desired units. In our rate conversion example, we are left with inches per hour, which may later be converted to another unit like feet per hour using the conversion factor \(1 \text{ ft} = 12 \text{ in.}\).

Dimensional Analysis

Dimensional analysis, also known as factor-label method or the unit-factor method, is a powerful tool used to convert one set of units to another, to check the validity of equations, or to solve quantitative problems. This technique revolves around the use of conversion factors and the arrangement of these factors to cancel out unwanted units and leave the desired units in the solution.

In our exercise \(27.5 \text{ in. } / 2.0 \mathrm{~h}= ?\), dimensional analysis ensures the proper units are in the numerator and the denominator, so the resulting rate correctly represents the physical quantity of interest. The clear and careful arrangement of units helps prevent errors in problem solving, making this method a cornerstone of successful chemistry problem-solving strategies.

Problem Solving in Chemistry

When tackling chemistry problems, problem-solving skills are of the utmost importance. Understanding the underlying principles at work and applying a systematic approach can often make complex problems manageable. The key steps typically involve identifying the given data, determining the unknown, selecting and applying the right concept or formula, and finally, carrying out the necessary mathematical operations.

In the context of the provided exercise, the initial steps involved identifying the given values (27.5 inches and 2.0 hours), then setting up the calculation properly, which in this case was a division, and proceeding to divide the distance by the time to find the rate. The final step involves providing the answer with the correct units. Educators often suggest that students write out all steps and keep a close eye on units throughout the process, as this is where many errors occur. Problem solving in chemistry relies heavily on a methodical approach and attention to detail, which helps avoid common mistakes and leads to more accurate and reliable results.