# Solving Real Problems With Chemistry

The early 19th-century discovery of the relationship between a gas's volume and temperature suggested that the volume of a gas would be zero at −273.15 °C.In 1848, British physicist William Thompson, who later adopted the title of Lord Kelvin, proposed an absolute temperature scale based on this concept (further treatment of this topic is provided in this text’s chapter on gases).

Note that, just as for numbers, when a unit is divided by an identical unit (in this case, m/m), the result is “1”—or, as commonly phrased, the units “cancel.” These calculations are examples of a versatile mathematical approach known as . are equivalent (by definition), and so a unit conversion factor may be derived from the ratio, $\mathrm$ Several other commonly used conversion factors are given in Table $$\Page Index$$.

When we multiply a quantity (such as distance given in inches) by an appropriate unit conversion factor, we convert the quantity to an equivalent value with different units (such as distance in centimeters).

To mark a scale on a thermometer, we need a set of reference values: Two of the most commonly used are the freezing and boiling temperatures of water at a specified atmospheric pressure.

On the Celsius scale, 0 °C is defined as the freezing temperature of water and 100 °C as the boiling temperature of water.

The space between the two temperatures is divided into 100 equal intervals, which we call degrees.

On the scale, the freezing point of water is defined as 32 °F and the boiling temperature as 212 °F.The space between these two points on a Fahrenheit thermometer is divided into 180 equal parts (degrees).Defining the Celsius and Fahrenheit temperature scales as described in the previous paragraph results in a slightly more complex relationship between temperature values on these two scales than for different units of measure for other properties.Representing the Celsius temperature as $$x$$ and the Fahrenheit temperature as $$y$$, the slope, $$m$$, is computed to be: $\begin m &=\dfrac \[4pt] &= \mathrm \[4pt] &= \mathrm \[4pt] &= \mathrm\end$ The y-intercept of the equation, , is then calculated using either of the equivalent temperature pairs, (100 °C, 212 °F) or (0 °C, 32 °F), as: $\begin b&=y-mx \[4pt] &= \mathrm \[4pt] &= \mathrm \end$ The equation relating the temperature scales is then: $\mathrm$ An abbreviated form of this equation that omits the measurement units is: $\mathrm$ Rearrangement of this equation yields the form useful for converting from Fahrenheit to Celsius: $\mathrm$ As mentioned earlier in this chapter, the SI unit of temperature is the kelvin (K).Unlike the Celsius and Fahrenheit scales, the kelvin scale is an absolute temperature scale in which 0 (zero) K corresponds to the lowest temperature that can theoretically be achieved.This is why it is referred to as the factor-label method.As your study of chemistry continues, you will encounter many opportunities to apply this approach.For example, a basketball player’s vertical jump of 34 inches can be converted to centimeters by: $\mathrm$ Since this simple arithmetic involves .The numbers of these two quantities are multiplied to yield the number of the product quantity, 86, whereas the units are multiplied to yield $\mathrm.$ Just as for numbers, a ratio of identical units is also numerically equal to one, $\mathrm$ and the unit product thus simplifies to .This is typically accomplished by measuring the from the equation that relates these three properties: $\mathrm$ An Olympic-quality sprinter can run 100 m in approximately 10 s, corresponding to an average speed of $\mathrm$ Note that this simple arithmetic involves dividing the numbers of each measured quantity to yield the number of the computed quantity (100/10 = 10) dividing the units of each measured quantity to yield the unit of the computed quantity (m/s = m/s).Now, consider using this same relation to predict the time required for a person running at this speed to travel a distance of 25 m.

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