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Chapter 13: Problem 24

The fuselage of an Airbus A340 has a circumference of \(17.72 \mathrm{m}\) on the ground. The circumference increases by \(26 \mathrm{cm}\) when it is in flight. Part of this increase is due to the pressure difference between the inside and outside of the plane and part is due to the increase in the temperature due to air drag while it is flying along at $950 \mathrm{km} / \mathrm{h} .$ Suppose we wanted to heat a full-size model of the airbus made of aluminum to cause the same increase in circumference without changing the pressure. What would be the increase in temperature needed?

Short Answer

Expert verified
Answer: The temperature increase needed is approximately 490.27 K.

Step by step solution

01

Convert the units for the change in circumference

First, convert the circumference increase from centimeters to meters for consistency in units: - \(26\,\text{cm} = 0.26\,\text{m}\) Now we have \(\Delta L = 0.26\,\mathrm{m}\).
02

Rearrange the formula for linear thermal expansion

Rearrange the formula for \(\Delta T\): \(\Delta T = \dfrac{\Delta L}{L_{0} \cdot \alpha}\) Plug in the given values: \(L_{0} = 17.72\,\mathrm{m}\) \(\Delta L = 0.26\,\mathrm{m}\) \(\alpha_{Al} = 24 \times 10^{-6} \mathrm{K^{-1}}\) Note that we are assuming the linear thermal expansion coefficient remains constant over the range of temperatures.
03

Calculate the increase in temperature

Substitute the values into the formula: \(\Delta T = \dfrac{0.26\,\mathrm{m}}{17.72\,\mathrm{m} \cdot (24 \times 10^{-6} \mathrm{K^{-1}})}\) \(\Delta T \approx 490.27\,\text{K}\) The increase in temperature needed to cause the same increase in circumference without changing the pressure is approximately \(490.27\,\text{K}\).

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Most popular questions from this chapter

Agnes Pockels \((1862-1935)\) was able to determine Avogadro's number using only a few household chemicals, in particular oleic acid, whose formula is \(\mathrm{C}_{18} \mathrm{H}_{34} \mathrm{O}_{2}\) (a) What is the molar mass of this acid? (b) The mass of one drop of oleic acid is \(2.3 \times 10^{-5} \mathrm{g}\) and the volume is $2.6 \times 10^{-5} \mathrm{cm}^{3} .$ How many moles of oleic acid are there in one drop? (c) Now all Pockels needed was to find the number of molecules of oleic acid. Luckily, when oleic acid is spread out on water, it lines up in a layer one molecule thick. If the base of the molecule of oleic acid is a square of side \(d\), the height of the molecule is known to be \(7 d .\) Pockels spread out one drop of oleic acid on some water, and measured the area to be \(70.0 \mathrm{cm}^{2}\) Using the volume and the area of oleic acid, what is \(d ?\) (d) If we assume that this film is one molecule thick, how many molecules of oleic acid are there in the drop? (e) What value does this give you for Avogadro's number?
Find the mass (in \(\mathrm{kg}\) ) of one molecule of \(\mathrm{CO}_{2}\)
A hydrogen balloon at Earth's surface has a volume of \(5.0 \mathrm{m}^{3}\) on a day when the temperature is \(27^{\circ} \mathrm{C}\) and the pressure is \(1.00 \times 10^{5} \mathrm{N} / \mathrm{m}^{2} .\) The balloon rises and expands as the pressure drops. What would the volume of the same number of moles of hydrogen be at an altitude of \(40 \mathrm{km}\) where the pressure is \(0.33 \times 10^{3} \mathrm{N} / \mathrm{m}^{2}\) and the temperature is \(-13^{\circ} \mathrm{C} ?\)
Estimate the mean free path of a \(\mathrm{N}_{2}\) molecule in air at (a) sea level \((P \approx 100 \mathrm{kPa} \text { and } T \approx 290 \mathrm{K}),\) (b) the top of \(\quad\) Mt. Everest (altitude $=8.8 \mathrm{km}, P \approx 50 \mathrm{kPa},\( and \)T \approx 230 \mathrm{K}),\( and \)(\mathrm{c})$ an altitude of \(30 \mathrm{km}(P \approx 1 \mathrm{kPa}\) and \(T=230 \mathrm{K}) .\) For simplicity, assume that air is pure nitrogen gas. The diameter of a \(\mathrm{N}_{2}\) molecule is approximately \(0.3 \mathrm{nm}\)
The temperature at which liquid nitrogen boils (at atmospheric pressure) is \(77 \mathrm{K}\). Express this temperature in (a) \(^{\circ} \mathrm{C}\) and (b) \(^{\circ} \mathrm{F}\).
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