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Chapter 2: Problem 13

In everyday experience, the measures of temperature most often used are Fahrenheit \(F\) and Celsius \(C\). Recall that the relationship between them is given by $$ F=1.8 C+32 $$ Physicists and chemists often use the Kelvin temperature scale. \({ }^{15}\) You can get kelvins \(K\) from degrees Celsius by using $$ K=C+273.15 . $$ a. Explain in practical terms what \(K\) (30) means, and then calculate that value. b. Find a formula expressing the temperature \(C\) in degrees Celsius as a function of the temperature \(K\) in kelvins. c. Find a formula expressing the temperature \(F\) in degrees Fahrenheit as a function of the temperature \(K\) in kelvins. d. What is the temperature in degrees Fahrenheit of an object that is 310 kelvins?

Short Answer

Expert verified
30 Kelvin is very cold, about -243.15 掳C. Formulas: \(C = K - 273.15\), \(F = 1.8K - 459.67\). 310 K is 98.33 掳F.

Step by step solution

01

Understanding Kelvin Temperature

In practical terms, a Kelvin temperature is the absolute temperature scale, starting at absolute zero, the point where all molecular motion stops. For instance, when the temperature is 30 K (30 Kelvin), it's 30 degrees above absolute zero, indicating a very cold condition compared to everyday temperature scales.
02

Calculate Value of Kelvin 30

To calculate 30 Kelvin in Celsius, we use the formula: \[ C = K - 273.15 \] Substituting the value, \[ C = 30 - 273.15 = -243.15 \] Thus, 30 K corresponds to -243.15 掳C.
03

Derive Celsius from Kelvin

To express Celsius as a function of Kelvin, use the given conversion formula:\[ C = K - 273.15 \]This equation directly translates temperatures from kelvin to degrees celsius.
04

Express Fahrenheit from Kelvin

To find a formula for Fahrenheit in terms of Kelvin, first convert Kelvin to Celsius, then use the Celsius to Fahrenheit formula:1. Use \( C = K - 273.15 \) to find Celsius.2. Plug into the Fahrenheit formula: \[ F = 1.8C + 32 \]Substitute to get: \[ F = 1.8(K - 273.15) + 32 \]Simplify to: \[ F = 1.8K - 459.67 \] Thus, this is the Fahrenheit-Kelvin relationship.
05

Calculate Fahrenheit for 310 Kelvin

Use the formula \( F = 1.8K - 459.67 \):\[ F = 1.8(310) - 459.67 \]\[ F = 558 - 459.67 = 98.33 \]Therefore, an object at 310 Kelvin is approximately 98.33 掳F.

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Key Concepts

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

Fahrenheit to Celsius
Temperature conversion from Fahrenheit to Celsius is one of the most common and necessary calculations in our daily lives. The relationship between these two temperature scales is succinctly given by the formula: \[ F = 1.8C + 32 \] This equation lets you easily convert a temperature in Celsius to its equivalent in Fahrenheit. To convert Fahrenheit to Celsius, however, you would rearrange this formula to solve for \( C \): \[ C = \frac{F - 32}{1.8} \] This subtraction and division will give you the Celsius temperature reading when you're starting with a Fahrenheit measurement. It's useful in everyday scenarios like checking weather forecasts or cooking with recipes from different countries.
Also, a few key points to note:
  • Freezing point of water: 32掳F (0掳C)
  • Boiling point of water: 212掳F (100掳C)
Celsius to Kelvin
The transition from Celsius to Kelvin is critical for scientific calculations, especially in chemistry and physics. The Kelvin scale is an absolute scale starting at absolute zero, where all molecular motion ceases. The conversion formula from Celsius to Kelvin is simple: \[ K = C + 273.15 \] It's all about adding 273.15 to your Celsius temperature. This results in a Kelvin value that conveniently avoids negative numbers, facilitating scientific measurement. Kelvin is the SI unit for temperature and useful for equations that require absolutes like energy calculations.
Some key facts about the Kelvin scale:
  • Absolute zero: 0 K
  • Freezing point of water: 273.15 K
  • Boiling point of water: 373.15 K
Kelvin to Fahrenheit
Converting Kelvin to Fahrenheit involves a two-step process because you need to pass through the Celsius scale first. It sounds complicated, but with formulas, it becomes straightforward. First, convert Kelvin to Celsius: \[ C = K - 273.15 \] Then, use the Celsius to Fahrenheit conversion formula: \[ F = 1.8C + 32 \] Combining these, you get the streamlined formula: \[ F = 1.8K - 459.67 \] This equation directly converts any Kelvin temperature to Fahrenheit, which is particularly useful when you need to explain temperatures in terms that are more familiar in daily life. For instance, converting Kelvin to Fahrenheit can describe scientific observations in a more relatable temperature scale.
Temperature Scales
Understanding different temperature scales is essential, especially in global and scientific contexts. The three scales you'll encounter most are Fahrenheit, Celsius, and Kelvin. Each scale serves unique purposes:
  • Fahrenheit: Commonly used in the United States for weather and cooking temperatures. Water freezes at 32掳F and boils at 212掳F.
  • Celsius: Widely used worldwide and in scientific settings. Water freezes at 0掳C and boils at 100掳C. This scale is often preferred due to its simplicity.
  • Kelvin: Essential for scientific research because it starts at absolute zero. It does not use degrees and is a direct measure of thermal motion in particles.
In practical terms, each scale's use is context-dependent. Fahrenheit provides practicality and ease in environments familiar with it, while Celsius is simple and aligns with metric measurements. Kelvin provides an absolute context pivotal for scientific exploration, such as in physics and chemistry. Understanding these scales empowers students and professionals to perform accurate and meaningful temperature-related calculations.

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Most popular questions from this chapter

The weekly profit \(P\) for a widget producer is a function of the number \(n\) of widgets sold. The formula is $$ P=-2+2.9 n-0.3 n^{2} $$ Here \(P\) is measured in thousands of dollars, \(n\) is measured in thousands of widgets, and the formula is valid up to a level of 7 thousand widgets sold. a. Make a graph of \(P\) versus \(n\). b. Calculate \(P(0)\) and explain in practical terms what your answer means. c. At what sales level is the profit as large as possible?

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For retailers who buy from a distributor or manufacturer and sell to the public, a major concern is the cost of maintaining unsold inventory. You must have appropriate stock to do business, but if you order too much at a time, your profits may be eaten up by storage costs. One of the simplest tools for analysis of inventory costs is the basic or der quantity model. It gives the yearly inventory expense \(E=E(c, N, Q, f)\) when the following inventory and restocking cost factors are taken into account: \- The carrying cost \(c\), which is the cost in dollars per year of keeping a single unsold item in your warehouse. \- The number \(N\) of this item that you expect to sell in 1 year. \- The number \(Q\) of items you order at a time. \- The fixed costs \(f\) in dollars of processing a restocking order to the manufacturer. (Note: This is not the cost of the order; the price of an item does not play a role here. Rather, \(f\) is the cost you would incur with any order of any size. It might include the cost of processing the paperwork, fixed costs you pay the manufacturer for each order, shipping charges that do not depend on the size of the order, the cost of counting your inventory, or the cost of cleaning and rearranging your warehouse in preparation for delivery.) The relationship is given by $$ E=\left(\frac{Q}{2}\right) c+\left(\frac{N}{Q}\right) f \text { dollars per year. } $$ A new-car dealer expects to sell 36 of a particular model car in the next year. It costs \(\$ 850\) per year to keep an unsold car on the lot. Fixed costs associated with preparing, processing, and receiving a single order from Detroit total \(\$ 230\) per order. a. Using the information provided, express the yearly inventory expense \(E=E(Q)\) as a function of \(Q\), the number of automobiles included in a single order. b. What is the yearly inventory expense if 3 cars at a time are ordered? c. How many cars at a time should be ordered to make yearly inventory expenses a minimum? d. Using the value of \(Q\) you found in part c, determine how many orders to Detroit will be placed this year. e. What is the average rate of increase in yearly inventory expense from the number you found in part \(\mathrm{c}\) to an order of 2 cars more?

If you borrow \(\$ 120,000\) at an APR of \(6 \%\) in order to buy a home, and if the lending institution compounds interest continuously, then your monthly payment \(M=M(Y)\), in dollars, depends on the number of years \(Y\) you take to pay off the loan. The relationship is given by $$ M=\frac{120000\left(e^{0.005}-1\right)}{1-e^{-0.06 Y}}. $$ a. Make a graph of \(M\) versus \(Y\). In choosing a graphing window, you should note that a home mortgage rarely extends beyond 30 years. b. Express in functional notation your monthly payment if you pay off the loan in 20 years, and then use the graph to find that value. c. Use the graph to find your monthly payment if you pay off the loan in 30 years. d. From part b to part \(\mathrm{c}\) of this problem, you increased the debt period by \(50 \%\). Did this decrease your monthly payment by \(50 \%\) ? e. Is the graph concave up or concave down? Explain your answer in practical terms. f. Calculate the average decrease per year in your monthly payment from a loan period of 25 to a loan period of 30 years.

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