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Chapter 1: Problem 1

The Fahrenheit temperature scale is defined so that ice melts at\(32^{\circ} \mathrm{F}\) and water boils at \(212^{\circ} \mathrm{F}\) (a) Derive the formulas for converting from Fahrenheit to Celsius and back. (b) What is absolute zero on the Fahrenheit scale?

Short Answer

Expert verified
(a) Fahrenheit to Celsius: \( C = \frac{5}{9}(F - 32) \), Celsius to Fahrenheit: \( F = \frac{9}{5}C + 32 \). (b) Absolute zero is $-459.67^{ ext{掳F}}$.

Step by step solution

01

Understand the Relationship between Fahrenheit and Celsius

The Celsius temperature scale is defined with the freezing point at $0^{ ext{掳C}}$ and the boiling point at $100^{ ext{掳C}}$. Meanwhile, the Fahrenheit scale defines ice melting at $32^{ ext{掳F}}$ and boiling at $212^{ ext{掳F}}$. We use this information to derive the conversion formulas.
02

Establish the Formula for Fahrenheit to Celsius

To find the conversion formula, observe that for a \(100^{ ext{掳C}}\) change in Celsius, it corresponds to a \(180^{ ext{掳F}}\) change in Fahrenheit. Thus, the ratio of change is: \[ \frac{180}{100} = \frac{9}{5} \]. The formula for conversion is then derived as: \[ C = \frac{5}{9}(F - 32) \].
03

Establish the Formula for Celsius to Fahrenheit

We can rearrange the formula found in step 2 to express \(F\) in terms of \(C\): starting from \( C = \frac{5}{9}(F - 32) \), solve for \(F\): \[ F = \frac{9}{5}C + 32 \]. This gives us the conversion from Celsius to Fahrenheit.
04

Convert Absolute Zero to Fahrenheit

Absolute zero is \(-273.15^{ ext{掳C}}\). Using the formula for Celsius to Fahrenheit, substitute \(C = -273.15\): \[ F = \frac{9}{5}(-273.15) + 32 \], which calculates to approximately \(-459.67^{ ext{掳F}}\).

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Key Concepts

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

Temperature Conversion Formulas
Temperature conversion is an important concept for understanding how different temperature scales relate to each other. To convert temperatures between Celsius and Fahrenheit, we first need to understand their fundamental characteristics.
  • The Celsius scale is defined such that ice melts at 0掳C and water boils at 100掳C.
  • The Fahrenheit scale has ice melting at 32掳F and water boiling at 212掳F.
To derive the formula for converting Fahrenheit to Celsius, consider the difference in boiling and freezing points:
  • The change from freezing to boiling on the Celsius scale is 100 degrees.
  • On the Fahrenheit scale, it is 180 degrees.
Thus, the ratio of Fahrenheit to Celsius change is 9/5. Using this ratio, the formula for conversion from Fahrenheit to Celsius is: \[ C = \frac{5}{9}(F - 32) \]
Conversely, to derive the formula from Celsius to Fahrenheit, simply rearrange the equation, which results in:
\[ F = \frac{9}{5}C + 32 \]
These formulas allow straightforward conversions between the two temperature scales.
Celsius and Fahrenheit Scales
The Celsius and Fahrenheit scales are the most commonly used temperature scales today. Having a solid grasp of these scales is essential for scientific, academic, and practical applications.
  • The Celsius scale is part of the metric system and is widely used around the world for most temperature measurements.
  • The Fahrenheit scale, meanwhile, is primarily used in the United States for weather forecasts and many other non-scientific applications.
Both scales define temperature by setting two fixed points:
  • The freezing point of water is 0掳C or 32掳F.
  • The boiling point of water is set at 100掳C or 212掳F.
This relationship highlights that the scales are offset from each other by 32 degrees and differ in step changes. Understanding their definitions and connections helps in converting between the two effectively, ensuring accuracy in both domestic and professional settings.
Absolute Zero Temperature
Absolute zero is a fundamental concept in thermodynamics, marking the lowest thermodynamic temperature possible. It represents the point where particles have minimal energy and entropy.
In Celsius, absolute zero is at \[-273.15^{\circ}C \]. To find the equivalent in Fahrenheit, apply the Celsius to Fahrenheit conversion formula:\[ F = \frac{9}{5}(-273.15) + 32 \].
This calculation results in \[-459.67^{\circ}F \]. This extreme temperature serves as the zero point for the Kelvin scale as well, where it is simply 0 K.
Understanding absolute zero is crucial for fields like physics and chemistry, as it provides insight into the behavior of gases and other states of matter at extremely low temperatures. Moreover, it is important for understanding real-world phenomena such as superconductivity and the behavior of matter at the edge of human-created environments.

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Most popular questions from this chapter

By applying Newton's laws to the oscillations of a continuous medium, one can show that the speed of a sound wave is given by $$c_{s}=\sqrt{\frac{B}{\rho}}$$ where \(\rho\) is the density of the medium (mass per unit volume) and \(B\) is the bulk modulus, a measure of the medium's stiffness. More precisely, if we imagine applying an increase in pressure \(\Delta P\) to a chunk of the material, and this increase results in a (negative) change in volume \(\Delta V\), then \(B\) is defined as the change in pressure divided by the magnitude of the fractional change in volume: $$B \equiv \frac{\Delta P}{-\Delta V / V}$$ This definition is still ambiguous, however, because I haven't said whether the compression is to take place isothermally or adiabatically (or in some other way). (a) Compute the bulk modulus of an ideal gas, in terms of its pressure \(P,\) for both isothermal and adiabatic compressions. (b) Argue that for purposes of computing the speed of a sound wave, the adiabatic \(B\) is the one we should use. (c) Derive an expression for the speed of sound in an ideal gas, in terms of its temperature and average molecular mass. Compare your result to the formula for the rms speed of the molecules in the gas. Evaluate the speed of sound numerically for air at room temperature. (d) When Scotland's Battlefield Band played in Utah, one musician remarked that the high altitude threw their bagpipes out of tune. Would you expect altitude to affect the speed of sound (and hence the frequencies of the standing waves in the pipes)? If so, in which direction? If not, why not?

A 60-kg hiker wishes to climb to the summit of Mt. Ogden, an ascent of 5000 vertical feet \((1500 \mathrm{m})\) (a) Assuming that she is \(25 \%\) efficient at converting chemical energy from food into mechanical work, and that essentially all the mechanical work is used to climb vertically, roughly how many bowls of corn flakes (standard serving size 1 ounce, 100 kilocalories) should the hiker eat before setting out? (b) As the hiker climbs the mountain, three-quarters of the energy from the corn flakes is converted to thermal energy. If there were no way to dissipate this energy, by how many degrees would her body temperature increase? (c) In fact, the extra energy does not warm the hiker's body significantly; instead, it goes (mostly) into evaporating water from her skin. How many liters of water should she drink during the hike to replace the lost fluids? (At \(25^{\circ} \mathrm{C},\) a reasonable temperature to assume, the latent heat of vaporization of water is \(580 \mathrm{cal} / \mathrm{g}, 8 \%\) more than at \(100^{\circ} \mathrm{C} .\) )

Estimate how long it should take to bring a cup of water to boiling temperature in a typical 600 -watt microwave oven, assuming that all the energy ends up in the water. (Assume any reasonable initial temperature for the water.) Explain why no heat is involved in this process.

Calculate the total thermal energy in a gram of lead at room temperature, assuming that none of the degrees of freedom are "frozen out" (this happens to be a good assumption in this case).

Consider a uniform rod of material whose temperature varies only along its length, in the \(x\) direction. By considering the heat flowing from both directions into a small segment of length \(\Delta x,\) derive the heat equation, $$ \frac{\partial T}{\partial t}=K \frac{\partial^{2} T}{\partial x^{2}} $$ where \(K=k_{t} / c \rho, c\) is the specific heat of the material, and \(\rho\) is its density. (Assume that the only motion of energy is heat conduction within the rod; no energy enters or leaves along the sides.) Assuming that \(K\) is independent of temperature, show that a solution of the heat equation is $$ T(x, t)=T_{0}+\frac{A}{\sqrt{t}} e^{-x^{2} / 4 K t} $$ where \(T_{0}\) is a constant background temperature and \(A\) is any constant. Sketch (or use a computer to plot) this solution as a function of \(x,\) for several values of \(t\) Interpret this solution physically, and discuss in some detail how energy spreads through the rod as time passes.
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