Section 3.9: Measures of Temperature
- Page ID
- 222496
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)
\( \newcommand{\dsum}{\displaystyle\sum\limits} \)
\( \newcommand{\dint}{\displaystyle\int\limits} \)
\( \newcommand{\dlim}{\displaystyle\lim\limits} \)
\( \newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\)
( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\)
\( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)
\( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\)
\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)
\( \newcommand{\Span}{\mathrm{span}}\)
\( \newcommand{\id}{\mathrm{id}}\)
\( \newcommand{\Span}{\mathrm{span}}\)
\( \newcommand{\kernel}{\mathrm{null}\,}\)
\( \newcommand{\range}{\mathrm{range}\,}\)
\( \newcommand{\RealPart}{\mathrm{Re}}\)
\( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)
\( \newcommand{\Argument}{\mathrm{Arg}}\)
\( \newcommand{\norm}[1]{\| #1 \|}\)
\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)
\( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\AA}{\unicode[.8,0]{x212B}}\)
\( \newcommand{\vectorA}[1]{\vec{#1}} % arrow\)
\( \newcommand{\vectorAt}[1]{\vec{\text{#1}}} % arrow\)
\( \newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vectorC}[1]{\textbf{#1}} \)
\( \newcommand{\vectorD}[1]{\overrightarrow{#1}} \)
\( \newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}} \)
\( \newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}} \)
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\(\newcommand{\longvect}{\overrightarrow}\)
\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)
\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)- Convert measurements of temperature in different systems
Up until this point, we have worked with both the metric system and the U.S. customary system, converting between units of length, mass, capacity, area, and volume. Many of these conversions involved multiplying or dividing by known conversion factors. While the metric system often follows patterns based on powers of ten, U.S. customary conversions required memorizing relationships between units.
Temperature, however, is different from all the measurements we have studied so far. Instead of measuring size or quantity, temperature measures the degree of heat or cold. Because of this, temperature scales are based on reference points, not just unit size, and conversions between them cannot be done using multiplication or division alone.
There are three common temperature scales:
- Celsius (°C)
- Fahrenheit (°F)
- Kelvin (K).
The Celsius scale, which is considered metric, is widely used and is based on the freezing point (0°C) and boiling point (100°C) of water.
The Fahrenheit scale, used primarily in the United States, places the freezing point at 32°F and the boiling point at 212°F.
The Kelvin scale is used globally in scientific settings and begins at absolute zero, the lowest and coldest possible temperature, where molecular motion ceases. Kelvin is designed to measure absolute thermodynamic temperature, not the relative temperatures (like daily weather) often measured in Fahrenheit. Measures in Kelvin does not use "degrees" or the degree symbol like Celsius and Fahrenheit does.
To convert between these scales, we must use formulas that account for both differences in scale and differences in starting points. This makes temperature conversion a unique and fitting conclusion to our study of measurement, highlighting that not all quantities can be converted in the same way.
To convert a given measure in Celsius to Fahrenheit, use the formula
\[ F=\frac{9}{5}C+32\nonumber \]
where \(C\) is the given measure in Celsius and \(F\) is the temperature in Fahrenheit.
Using algebra, we can solve the equation for \(C\) in terms of \(F\), allowing us to derive a formula for converting in the opposite direction.
To convert a given measure in Fahrenheit to Celsius, use the formula
\[ C=\frac{5}{9}(F-32)\nonumber \]
where \(F\) is the temperature in Fahrenheit and \(C\) is the given measure in Celsius
Although Kelvin is not commonly used in everyday situations, the same types of formulas can still be applied to convert between temperature scales.
Here are the standard temperature conversion formulas between Kelvin and Celsius, and Fahrenheit:
\[ K=C+273.15\nonumber \]
\[ C=K-273.15\nonumber \]
where \(K\) is the temperature in Kelvin and \(C\) is the given measure in Celsius.
Here are the standard temperature conversion formulas between Kelvin and Fahrenheit:
\[ F=\frac{9}{5}(K-273.15)+32\nonumber \]
\[ K=\frac{5}{9}(F-32)+273.15\nonumber \]
where \(K\) is the temperature in Kelvin and \(F\) is the given measure in Fahrenheit.
Convert 95°F to Celsius.
✅ Solution:
Use the formula converting Farenheit to Celsius:
\[ C = \frac{5}{9}(F-32)\nonumber \]
Let \(F=98\) and solve for \(C\).
\[\begin{align} C &=\frac{5}{9}(95-32) \nonumber \\[8px] C &=\frac{5}{9}(63) \nonumber \\[8px] \end{align}\]
\[\boxed{C =35} \nonumber\]
So, 95°F is equivalent 35°C.
Convert 20°C to Farenheit.
✅ Solution:
Use the formula converting Celsius to Farenheit:
\[ F=\frac{9}{5}C+32\nonumber \]
Let \(C=20\) and solve for \(F\).
\[\begin{align} F &=\frac{9}{5}(20)+32 \nonumber \\[8px] F &=36+32 \nonumber \\[8px] \end{align}\]
\[\boxed{F=20} \nonumber\]
So, 20°C is equivalent 68°F.
Convert 46°F to Celsius.
✅ Solution:
Use the formula converting Farenheit to Celsius:
\[ C = \frac{5}{9}(F-32)\nonumber \]
Let \(F=46\) and solve for \(C\).
\[\begin{align} C &=\frac{5}{9}(46-32) \nonumber \\[8px] C &=\frac{5}{9}(14) \nonumber \\[8px] C &=7.7777777778 \nonumber \\[8px] \nonumber \end{align}\]
\[\boxed{C\approx 7.8} \nonumber\]
So, 46°F is equivalent 7.8°C.
Convert \(-8^{\circ}\)C to Farenheit.
✅ Solution:
Use the formula converting Celsius to Farenheit:
\[ F=\frac{9}{5}C+32\nonumber \]
Let \(C=-8\) and solve for \(F\).
\[\begin{align} F &=\frac{9}{5}(-8)+32 \nonumber \\[8px] F &=-14.4+32 \nonumber \\[8px] \end{align}\]
\[\boxed{F = 17.6} \nonumber\]
So, \(-\)8°C is equivalent 17.6°F.
Color temperature describes the color of light emitted by an object, and it is measured in 
Kelvin (K) rather than Celsius or Fahrenheit. Although it uses the word temperature, it does not directly measure how hot or cold something feels. Instead, it refers to the color that an ideal heated object would emit at a given temperature.
As an object is heated, it changes color in a predictable way:
- At lower temperatures, it glows reddish
- As the temperature increases, it becomes orange, yellow, and white
- At very high temperatures, it appears bluish
This relationship comes from physics and is based on how a “blackbody” radiates light.
As you can see in the figure to the right, some LED light bulbs offer a range of color temperatures, from warm amber tones to bright white. Convert the 4,000K that is labeled as "cool white" to Farenheit.
✅ Solution:
Use the formula converting Kelvin to Farenheit:
\[ F=\frac{9}{5}(K-273.15)+32\nonumber \]
Let \(K=\text{4,000}\) and solve for \(F\).
\[\begin{align} F &=\frac{9}{5}(\text{4,000}-273.15)+32 \nonumber \\[8px] F &=\frac{9}{5}(\text{3,726.85})+32 \nonumber \\[8px] F &=\frac{9}{5}(\text{6,708.33})+32 \nonumber \\[8px] F &=\text{6,740.33} \nonumber \\[8px] \nonumber \end{align}\]
\[\boxed{F\approx\text{6,740}} \nonumber\]
So, 4,000K is equivalent 6,740°F.
Note: Although Kelvin can be converted to Celsius or Fahrenheit, the temperatures involved are extremely high, so color temperature is almost always expressed in Kelvin rather than everyday temperature scales.
A common room temperature in science is approximately 80°F. What is the temperature in Kelvin?
✅ Solution:
Use the formula converting Kelvin to Farenheit:
\[ K=\frac{5}{9}(F-32)+273.15\nonumber \]
Let \(F=80\) and solve for \(K\).
\[\begin{align} K &=\frac{5}{9}(80-32)+273.15 \nonumber \\[8px] K &=\frac{5}{9}(48)+273.15 \nonumber \\[8px] K &=26.66666666667+273.15 \nonumber \\[8px] K &=299.816666667 \nonumber \\[8px] \nonumber \end{align}\]
\[\boxed{K\approx300} \nonumber\]
So, 4,000°F is equivalent 300K.
-
The highest reliably recorded air temperature on Earth occurred in Furnace Creek, Death Valley National Park on July 10th, 1913 at 134.1°F. Convert this temperature to Celsius. Round to the nearest tenths place.
-
Greenland is largely covered by an ice sheet (about 80% of it), making it one of the coldest and iciest places on Earth. Iceland, on the other hand, has green valleys, grasslands, and a milder climate, especially along the coasts. In a typical summer, Nuuk, the capital of Greenland, has an average temperature of 6.1°C. Convert this temperature to Farenheit. Round to the nearest tenths place.
-
Refer to the figure to the right. Convert the 3,000K that is labeled as "warm white" to Farenheit. Round to the nearest whole number.
- Answers
-
- 56.7°C
- 43.0°F
- 4,940K