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Chapter 10: Problem 8

The temperature difference between the inside and the outside of a home on a cold winter day is \(57.0^{\circ} \mathrm{F}\). Express this difference on (a) the Celsius scale and (b) the Kelvin scale.

Short Answer

Expert verified
The temperature difference is approximately \(13.89^{\circ} \mathrm{C}\) on the Celsius scale and \(287.04 \, \mathrm{K}\) on the Kelvin scale.

Step by step solution

01

Conversion to Celsius Scale

First, we convert the temperature difference from Fahrenheit to Celsius using the conversion formula \( C = (F - 32) * \frac{5}{9} \). Substituting \( F = 57.0^{\circ} \mathrm{F} \)into the formula gives \( C = (57.0 - 32) * \frac{5}{9} \). Calculating this yields a result of about \( 13.89^{\circ} \mathrm{C} \).
02

Conversion to Kelvin Scale

Next, convert the obtained Celsius temperature to Kelvin by adding 273.15 to the Celsius temperature. The Kelvin temperature is thus \( K = C + 273.15 \). Substituting the previously obtained Celsius temperature, we get \( K = 13.89 + 273.15 \) which results in approximately \( K = 287.04 \) K.

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Key Concepts

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

Celsius Scale
The Celsius scale is a widely used method for measuring temperatures, particularly in most countries around the world. It is based on the metric system, which means it's logically aligned with units like meters and grams.
In Celsius, water freezes at 0 degrees and boils at 100 degrees under standard atmospheric conditions. This temperature scale is particularly user-friendly because of its straightforward nature 鈥� it divides the temperature between freezing and boiling into 100 equal parts.
  • Freezing point: 0掳C
  • Boiling point: 100掳C
  • Common for science and international use
To convert a temperature from Fahrenheit to Celsius, use the formula:
\[C = (F - 32) \times \frac{5}{9}\]Where:
  • \( C \) is the temperature in degrees Celsius,
  • \( F \) is the temperature in degrees Fahrenheit.
Fahrenheit Scale
The Fahrenheit scale is primarily used in the United States and few other countries for non-scientific temperature measurement. On this scale, the freezing point of water is at 32 degrees and the boiling point is at 212 degrees.
It's named after the German physicist Daniel Gabriel Fahrenheit, who was instrumental in its development.
  • Freezing point: 32掳F
  • Boiling point: 212掳F
  • Common in household and weather temperatures in the US
The conversion from Fahrenheit to Celsius requires subtracting 32 from the Fahrenheit temperature, then multiplying by \( \frac{5}{9} \).
This step is essential when translating temperatures to make them meaningful in scientific contexts where Celsius or Kelvin are standard.
Kelvin Scale
The Kelvin scale is an absolute temperature scale mainly used in scientific contexts. Unlike Celsius and Fahrenheit, it does not use degrees but rather "kelvins." It starts from absolute zero, which is the theoretical lowest possible temperature where molecular motion virtually ceases.
In the Kelvin scale, water freezes at 273.15 K and boils at 373.15 K.
  • Absolute zero: 0 K
  • Freezing point of water: 273.15 K
  • Boiling point of water: 373.15 K
  • Common in physics and scientific research
To convert a temperature from Celsius to Kelvin, simply add 273.15 to the Celsius measurement:
\[K = C + 273.15\]Where:
  • \( K \) is the temperature in kelvins,
  • \( C \) is the temperature in degrees Celsius.

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Most popular questions from this chapter

(a) An ideal gas occupies a volume of \(1.0 \mathrm{~cm}^{3}\) at \(20^{\circ} \mathrm{C}\) and atmospheric pressure. Determine the number of molecules of gas in the container. (b) If the pressure of the \(1.0-\mathrm{cm}^{3}\) volume is reduced to \(1.0 \times 10^{-11}\) Pa (an extremely good vacuum) while the temperature remains constanL. how many moles of gas remain in the container?

A bimetallic bar is made of two thin strips of dissimilar metals bonded together. As they are heated, the one with the larger average coefficient of expansion expands more than the other, forcing the bar into an arc, with the outer strip having both a larger radius and a larger circumference (Fig. P10.61). (a) Derive an expression for the angle of bending, \(\theta\), as a function of the initial length of the strips. their average coefficientsof linear expansion, the change in temperature, and the separation of the centers of the strips \(\left(\Delta r=r_{2}-r_{1}\right) .\) (b) Show that the angle of bending goes to zero when \(\Delta T\) goes to zero or when the two coefficients of expansion become equal. (c) What happens if the bar is cooled?

Show that the coefficient of volume expansion, \(\beta\), is related to the coefficient of linear expansion, \(\alpha\), through the expression \(\beta=3 \alpha\)

Show that the temperature \(-40^{\circ}\) is unique in that it has the same numerical value on the Celsius and Fahrenheit scales.

A brass ring of diameter \(10.00 \mathrm{~cm}\) at \(20.0^{\circ} \mathrm{C}\) is heated and slipped over an aluminum rod of diameter \(10.01 \mathrm{~cm}\) at \(20.0^{\circ} \mathrm{C}\). Assuming the average coefficients of linear expansion are constant, (a) to what temperature must the combination be cooled to separate the two metals? Is that temperature attainable? (b) What if the aluminum rod were \(10.02 \mathrm{~cm}\) in diameter?
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