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Chapter 17: Problem 4

(a) Calculate the one temperature at which Fahrenheit and Celsius thermometers agree with each other. (b) Calculate the one temperature at which Fahrenheit and Kelvin thermometers agree with each other.

Short Answer

Expert verified
Fahrenheit and Celsius agree at -40 degrees; Fahrenheit and Kelvin agree at 574.59 degrees.

Step by step solution

01

Understanding the Fahrenheit-Celsius Relationship

The relationship between degrees Fahrenheit and degrees Celsius can be expressed with the formula \( F = \frac{9}{5}C + 32 \). Where \( F \) is the temperature in Fahrenheit, and \( C \) is the temperature in Celsius.
02

Setting Fahrenheit Equal to Celsius

To find the temperature where Fahrenheit and Celsius agree, set \( F = C \). Substitute \( C \) for \( F \) in the formula, resulting in \( C = \frac{9}{5}C + 32 \).
03

Solving the Equation for Celsius

Solve the equation \( C = \frac{9}{5}C + 32 \). Rearranging terms gives \( C - \frac{9}{5}C = 32 \). This simplifies to \(-\frac{4}{5}C = 32 \). Multiply both sides by \(-5/4\) to solve for \( C \):\[ C = 32 \times \left(-\frac{5}{4}\right) = -40 \]
04

Verifying the Solution

Convert \( -40 \) Celsius using the formula: \( F = \frac{9}{5}(-40) + 32 = -40 \). Both thermometers show \(-40\), confirming the solution.
05

Understanding the Fahrenheit-Kelvin Relationship

The conversion formula between Fahrenheit and Kelvin is \( F = \frac{9}{5}(K - 273.15) + 32 \). Where \( F \) is the degrees Fahrenheit and \( K \) is the temperature in Kelvin.
06

Setting Fahrenheit Equal to Kelvin

To find the temperature where Fahrenheit and Kelvin agree, set \( F = K \). Substitute \( K \) for \( F \) in the formula: \( K = \frac{9}{5}(K - 273.15) + 32 \).
07

Solving the Equation for Kelvin

Solve the equation \( K = \frac{9}{5}(K - 273.15) + 32 \). Distribute and rearrange:\[ K = \frac{9}{5}K - \frac{9}{5} \times 273.15 + 32 \]This simplifies to:\[ K - \frac{9}{5}K = - \frac{9 \times 273.15}{5} + 32 \]\[ -\frac{4}{5}K = -491.67 + 32 \]\[ -\frac{4}{5}K = -459.67 \]Multiply both sides by \(-5/4\) to solve for \( K \):\[ K = -459.67 \times \left(-\frac{5}{4}\right) = 574.59 \]
08

Verifying the Solution

Convert \(574.59 \) Kelvin using the formula: \( F = \frac{9}{5}(574.59 - 273.15) + 32 \) which simplifies to \(574.59 \) agreeing with the Kelvin value.

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Key Concepts

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

Fahrenheit and Celsius
The relationship between Fahrenheit and Celsius is fundamental to understanding temperature conversion. When you want to convert from Celsius to Fahrenheit, you use the formula:
  • \( F = \frac{9}{5}C + 32 \)
Here, \( F \) represents the temperature in degrees Fahrenheit and \( C \) represents the temperature in degrees Celsius.
To find the temperature where the two scales agree, meaning both thermometers show the same reading, we set the two equal:
  • \( F = C \)
This results in the equation:
  • \( C = \frac{9}{5}C + 32 \)
Solving this, we rearrange and simplify to find that \( C = -40 \).
At this temperature, \(-40\)掳C indeed converts to \(-40\)掳F when checked using the formula. Understanding this special intersection helps solidify grasp of how Fahrenheit and Celsius differ and align.
Fahrenheit and Kelvin
The relationship between Fahrenheit and Kelvin is slightly different and less intuitive than Celsius. The conversion formula is:
  • \( F = \frac{9}{5}(K - 273.15) + 32 \)
Where \( F \) is in Fahrenheit and \( K \) is in Kelvin.
To determine where Fahrenheit matches Kelvin exactly, set \( F = K \):
  • \( K = \frac{9}{5}(K - 273.15) + 32 \)
Solving for \( K \), you simplify the equation to eventually find that \( K = 574.59 \).
When you backtrack by converting \(574.59\) Kelvin to Fahrenheit using the original formula, it equals \(574.59\)掳F, verifying the equivalence. This intersection temperature reveals an interesting interplay between these systems, emphasizing the complexity of temperature conversion.
Temperature Equivalence
Temperature equivalence occurs when two different measures show the same number value for a specific point. Understanding this concept requires navigating through precise equations.
When solving problems of equivalence, first set the formulas equal to each other, given that they represent the same physical situation.
  • Fahrenheit and Celsius example: \(-40\) is the intersection point.
  • Fahrenheit and Kelvin take a different mathematical path, resulting in \(574.59\).
Finding these equivalent points encourages not only practicing algebra but also deepening one's grasp of how different systems are interconnected.
Having these skills is essential for students to solve real-world problems where temperature conversion is necessary across different scientific domains.

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Most popular questions from this chapter

A Styrofoam bucket of negligible mass contains 1.75 \(\mathrm{kg}\) of water and 0.450 \(\mathrm{kg}\) of ice. More ice, from a refrigerator at \(-15.0^{\circ} \mathrm{C},\) is added to the mixture in the bucket, and when thermal equilibrium has been reached, the total mass of ice in the bucket is 0.868 \(\mathrm{kg} .\) Assuming no heat exchange with the surroundings, what mass of ice was added?

One end of an insulated metal rod is maintained at \(100.0^{\circ} \mathrm{C},\) and the other end is maintained at \(0.00^{\circ} \mathrm{C}\) by an ice-water mixture. The rod is 60.0 \(\mathrm{cm}\) long and has a cross-sectional area of 1.25 \(\mathrm{cm}^{2} .\) The heat conducted by the rod melts 8.50 \(\mathrm{g}\) of ice in 10.0 \(\mathrm{min} .\) Find the thermal conductivity \(k\) of the metal.

A technician measures the specific heat of an unidentified liquid by immersing an electrical resistor in it. Electrical energy is converted to heat transferred to the liquid for 120 s at a constant rate of 65.0 W. The mass of the liquid is \(0.780 \mathrm{kg},\) and its temperature increases from \(18.55^{\circ} \mathrm{C}\) to \(22.54^{\circ} \mathrm{C}\) . (a) Find the average specific heat of the liquid in this temperature range. Assume that negligible heat is transferred to the container that holds the liquid and that no heat is lost to the surroundings. (b) Suppose that in this experiment heat transfer from the liquid to the container or surroundings cannot be ignored. Is the result calculated in part (a) an overestimate or an underestimate of the average specific heat? Explain.

A crate of fruit with mass 35.0 \(\mathrm{kg}\) and specific heat 3650 \(\mathrm{J} / \mathrm{kg} \cdot \mathrm{K}\) slides down a ramp inclined at \(36.9^{\circ}\) below the horizontal. The ramp is 8.00 \(\mathrm{m}\) long. (a) If the crate was at rest at the top of the incline and has a speed of 2.50 \(\mathrm{m} / \mathrm{s}\) at the bottom, how much work was done on the crate by friction? (b) If an amount of heat equal to the magnitude of the work done by friction goes into the crate of fruit and the fruit reaches a uniform final temperature, what is its temperature change?

Temperatures in Biomedicine. (a) Normal body temperature. The average normal body temperature measured in the mouth is 310 \(\mathrm{K}\) . What would Celsius and Fahrenheit thermometers read for this temperature? (b) Elevated body temperature. During very vigorous exercise, the body's temperature can go as high as \(40^{\circ} \mathrm{C} .\) What would Kelvin and Fahrenheit thermometers read for this temperature? (c) Temperature difference in the body. The surface temperature of the body is normally about 7 \(\mathrm{C}^{\circ}\) lower than the internal temperature. Express this temperature difference in kelvins and in Fahrenheit degrees. (d) Blood storage. Blood stored at \(4.0^{\circ} \mathrm{C}\) lasts safely for about 3 weeks, whereas blood stored at \(-160^{\circ} \mathrm{C}\) lasts for 5 years. Express both temperatures on the Fahrenheit and Kelvin scales. (e) Heat stroke. If the body's temperature is above \(105^{\circ} \mathrm{F}\) for a prolonged period, heat stroke can result. Express this temperature on the Celsius and Kelvin scales.
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