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Chapter 10: Problem 1

For each of the following temperatures, find the equivalent temperature on the indicated scale: (a) \(-273.15^{\circ} \mathrm{C}\) on the Fahrenheit scale, (b) \(98.6 \mathrm{~F}\) on the Celsius scale, and \((c) 100 \mathrm{~K}\) on the Fahrenheit scale.

Short Answer

Expert verified
(a) -459.67 degrees Fahrenheit (b) 37 degrees Celsius (c) -279.67 degrees Fahrenheit

Step by step solution

01

Converting Celsius to Fahrenheit

Formula to convert Celsius (\(C\)) to Fahrenheit (\(F\)) is given by \(F = C*1.8 + 32\). Substitute \(C = -273.15\) into this equation to obtain Fahrenheit.
02

Converting Fahrenheit to Celsius

Formula to convert Fahrenheit (\(F\)) to Celsius (\(C\)) is given by \(C = (F-32) / 1.8\). Substitute \(F = 98.6\) into this equation to obtain Celsius.
03

Converting Kelvin to Fahrenheit

First, convert Kelvin (\(K\)) to Celsius using formula \(C = K - 273.15\). Substitute \(K = 100\) into this equation to obtain Celsius. Then use the formula from step 1 to convert Celsius to Fahrenheit.

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Key Concepts

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

Celsius to Fahrenheit
When converting temperatures from Celsius to Fahrenheit, a simple formula helps you make the switch. The formula is: \(F = C \times 1.8 + 32\).
This equation allows you to calculate the temperature in Fahrenheit (\(F\)) once you know the temperature in Celsius (\(C\)).
For instance, if you need to convert \(-273.15^{\circ} \mathrm{C}\) (which is absolute zero) to Fahrenheit, plug the Celsius value into the equation like this: \(-273.15 \times 1.8 + 32\).
  • The multiplication \(-273.15 \times 1.8\) yields \(-491.67\).
  • Adding \(32\) results in a final Fahrenheit temperature of \(-459.67^{\circ} \mathrm{F}\).
This detailed step-by-step approach emphasizes the importance of using the formula accurately, especially with negative values.
Fahrenheit to Celsius
Converting Fahrenheit to Celsius involves rearranging the previous formula to calculate the Celsius temperature. The necessary formula is: \(C = \frac{F-32}{1.8}\).
This formula reverses the conversion, helping you find Celsius (\(C\)) from a given Fahrenheit value (\(F\)).
Let's consider the example where \(98.6^{\circ} \mathrm{F}\) needs conversion to Celsius. Following the formula: \((98.6 - 32) \div 1.8\).
  • Subtracting \(32\) from \(98.6\) results in \(66.6\).
  • Then, dividing \(66.6\) by \(1.8\) gives \(37^{\circ} \mathrm{C}\).
By following these steps, you turn a temperature from Fahrenheit to Celsius accurately, shown here with a common human body temperature example.
Kelvin to Fahrenheit
To convert a temperature from Kelvin to Fahrenheit, use a two-step process. First, convert Kelvin to Celsius, then Celsius to Fahrenheit.
Start by using the formula: \(C = K - 273.15\), where \(K\) is the Kelvin temperature.
Then, use the Celsius obtained to find the Fahrenheit equivalent.
Take, for example, \(100 \mathrm{~K}\). First, converting Kelvin to Celsius: \(100 - 273.15 = -173.15^{\circ} \mathrm{C}\).
  • Next, convert this Celsius value to Fahrenheit: \(-173.15 \times 1.8 + 32\).
  • The calculation \(-173.15 \times 1.8\) results in \(-311.67\).
  • Adding \(32\) gives you a result of \(-279.67^{\circ} \mathrm{F}\).
This comprehensive approach ensures you don't miss any steps and accurately find the Fahrenheit temperature from Kelvin.

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Most popular questions from this chapter

\(A\) bicycle tire is inflated to a gauge pressure of \(2.5 \mathrm{~atm}\) when the temperature is \(15^{\circ} \mathrm{C}\). While a man is riding the bicycle, the temperature of the tire increases to \(45^{\circ} \mathrm{C}\). Assuming the volume of the tire does not change, what is the gauge pressure in the tire at the higher temperature?

A vertical cylinder of crosssectional area \(A\) is fitted with \(\underline{a}\) tight-fitting, frictionless piston of mass \(m\) (Fig. P10.54). (a) If \(n\) moles of an ideal gas are in the cylinder at a temperature of \(T\), use Newton's second law for equilibrium to show that the height \(h\) at which the piston is in equilibrium under its own weight is given by $$ h=\frac{n R T}{m g+P_{n} A} $$ where \(P_{0}\) is atmospheric pressure. (b) Is the pressure inside the cylinder less than, equal to, or greater than atmospheric pressure? (c) If the gas in the cylinder is warmed, how would the answer for \(h\) be affected?

Two small containers, each with a volume of \(100 \mathrm{~cm}^{3}\), contain helium gas at \(0^{-} \mathrm{C}\) and \(1.00\) atm pressure. The two containers are joined by a small open tube of negligible volume, allowing gas to flow from one container to the other. What common pressure will exist in the two containers if the temperature of one container is raised to \(100^{\circ} \mathrm{C}\) while the other container is kept at \(0^{\circ} \mathrm{C}\) ?

What is the average kinetic energy of a molecule of oxygen at a temperature of \(300 \mathrm{~K}\) ?

Before beginning a long trip on a hot day, a driver inflates an automobile tire to a gauge pressure of \(1.80\) atm at \(300 \mathrm{~K}\). At the end of the trip, the gauge pressure has increased to \(2.20\) atm. (a) Assuming the volume has remained constant, what is the temperature of the air inside the tire? (b) What percentage of the original mass of air in the tire should be released so the pressure returns to its original value? Assume the temperature remains at the value found in part (a) and the volume of the tire remains constant as air is released.
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