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Chapter 16: Problem 12

At what temperature does the numerical value in \(^{\circ} \mathrm{F}\) match the numerical value in \(^{\circ} \mathrm{C} ?\)

Short Answer

Expert verified
The numerical value in Fahrenheit matches the numerical value in Celsius at -40 degrees.

Step by step solution

01

Set Up the Equality

Set up an equation to represent the situation where the degrees in Fahrenheit equals the degrees in Celsius, i.e., \(^{\circ} \mathrm{C} = ^{\circ} \mathrm{F}. Substituting the conversion formula into this equation yields: \(^{\circ} \mathrm{F}=(^{\circ} \mathrm{F} - 32) \times 5/9.
02

Simplify the Equation

Simplify the equation by distributing the right side: \(^{\circ} \mathrm{F} = 5/9 \times ^{\circ} \mathrm{F} - 160/9. Then, rearrange the equation so that all terms involving \(^{\circ} \mathrm{F} are on the same side, i.e., \(4/9 \times ^{\circ} \mathrm{F}= - 160/9.
03

Solve for Fahrenheit

Solve for \(^{\circ} \mathrm{F} by multiplying both sides of the equation by \(9/4 to isolate \(^{\circ} \mathrm{F} on one side. The solution to the equation gives us \(^{\circ} \mathrm{F}= -40.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Fahrenheit to Celsius Conversion
Understanding how to convert temperatures from Fahrenheit to Celsius is essential for scientific studies, cooking, and travel to countries using different temperature scales. The formula to convert Fahrenheit to Celsius is given by:
\[\begin{equation} ^\circ\mathrm{C} = \frac{5}{9}(^\circ\mathrm{F} - 32) \end{equation}\] To convert from Celsius to Fahrenheit, the formula is the reverse: \[\begin{equation} ^\circ\mathrm{F} = (^\circ\mathrm{C} \times \frac{9}{5}) + 32 \end{equation}\]
These formulas are based on the different scales where water freezes at 32掳F or 0掳C and boils at 212掳F or 100掳C under standard conditions. By understanding these formulas, students can easily tackle problems involving temperature conversion.
Thermodynamic Temperature
Thermodynamic temperature is a foundational concept in physics and chemistry that measures the average kinetic energy of particles in a substance. The temperature of an object determines how rapidly its particles are moving. The SI unit for thermodynamic temperature is the Kelvin (K), which is related to Celsius through the formula: \[\begin{equation} K = ^\circ\mathrm{C} + 273.15 \end{equation}\]
Kelvin is an absolute temperature scale, which means it starts at absolute zero, the theoretical point where particles have minimal kinetic energy. This scale is crucial in scientific research because it provides a uniform standard for temperature measurement, unlike Fahrenheit or Celsius, which are more common in everyday contexts.
Solving Algebraic Equations
Solving algebraic equations is a key skill in mathematics. The goal is to isolate the variable of interest on one side of the equation to find its value. Here are some steps one might follow:
  • Distribute any constants across variables.
  • Combine like terms on each side of the equation.
  • Use addition or subtraction to get all terms with the variable on one side of the equal sign.
  • Use multiplication or division to isolate the variable and solve for its value.
For example, in the exercise provided, the equation was simplified and rearranged to isolate the Fahrenheit temperature, showing that the temperatures in Fahrenheit and Celsius are numerically equivalent at -40掳. This process involved distributing, combining like terms, and manipulating the equation by multiplying both sides to solve for the Fahrenheit value.

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Most popular questions from this chapter

0.10 mol of argon gas is admitted to an evacuated \(50 \mathrm{cm}^{3}\) container at \(20^{\circ} \mathrm{C}\). The gas then undergoes an isobaric heating to a temperature of \(300^{\circ} \mathrm{C}\) a. What is the final volume of the gas? b. Show the process on a \(p V\) diagram. Include a proper scale on both axes.

A diving bell is a 3.0 -m-tall cylinder closed at the upper end but open at the lower end. The temperature of the air in the bell is \(20^{\circ} \mathrm{C} .\) The bell is lowered into the ocean until its lower end is \(100 \mathrm{m}\) deep. The temperature at that depth is \(10^{\circ} \mathrm{C}.\) a. How high does the water rise in the bell after enough time has passed for the air inside to reach thermal equilibrium? b. A compressed-air hose from the surface is used to expel all the water from the bell. What minimum air pressure is needed to do this?

\(10,000 \mathrm{cm}^{3}\) of \(200^{\circ} \mathrm{C}\) steam at a pressure of 20 atm is cooled until it condenses. What is the volume of the liquid water? Give your answer in \(\mathrm{cm}^{3}\).

The solar corona is a very hot atmosphere surrounding the visible surface of the sun. X-ray emissions from the corona show that its temperature is about \(2 \times 10^{6} \mathrm{K}\). The gas pressure in the corona is about \(0.03 \mathrm{Pa}\). Estimate the number density of particles in the solar corona.

The volume in a constant-pressure gas thermometer is directly proportional to the absolute temperature. A constant pressure thermometer is calibrated by adjusting its volume to \(1000 \mathrm{mL}\) while it is in contact with a reference cell at the triple point of water. The volume increases to 1638 mL when the thermometer is placed in contact with a sample. What is the sample's temperature?
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