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Chapter 10: Problem 5
Show that the temperature \(-40^{\circ}\) is unique in that it has the same numerical value on the Celsius and Fahrenheit scales.
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Long-term space missions require reclamation of the oxygen in the carbon dioxide exhaled by the crew. In one method of reclamation, \(1.00 \mathrm{~mol}\) of carbon dioxide produces \(1.00 \mathrm{~mol}\) of oxygen, with \(1.00 \mathrm{~mol}\) of methane as a by-product. The methane is stored in a tank under pressure and is available to control the attitude of the spacecraft by controlled venting. A single astronaut exhales \(1.09 \mathrm{~kg}\) of carbon dioxide each day. If the methane generated in the recycling of three astronauts' respiration during one week of flight is stored in an originally empty \(150-\mathrm{L}\) tank at \(-45.0^{\circ} \mathrm{C}\), what is the final pressure in the tank?
S Show that the coefficient of volume expansion \(\beta\), is related to the coefficient of linear expansion, \(\alpha\) through the expression \(\beta=3 \alpha\).
\(M\) The active element of a certain laser is made of a glass rod \(30.0 \mathrm{~cm}\) long and \(1.50 \mathrm{~cm}\) in diameter. Assume the average coefficient of linear expansion of the glass is \(9.00 \times 10^{-6}\left({ }^{\circ} \mathrm{C}\right)^{-1}\). If the temperature of the rod increases by \(65.0^{\circ} \mathrm{C}\), what is the increase in (a) its length, (b) its diameter, and (c) its volume?
A \(1.5\)-m-long glass tube that is closed at one end is weighted and lowered to the bottom of a freshwater lake. When the tube is recovered, an indicator mark shows that water rose to within \(0.40 \mathrm{~m}\) of the closed end. Determine the depth of the lake. Assume constant temperature.
A liquid with a coefficient of volume expan- sion of \(\beta\) just fills a spherical flask of volume \(V_{0}\) at tem- perature \(T_{i}\) (Fig. P10.57). The flask is made of a mate- rial that has a coefficient of linear expansion of \(\alpha\). The liquid is free to expand into a capillary of cross-sectional area \(A\) at the top. (a) Show that if the temperature increases by \(\Delta T\), the liquid rises in the capillary by the amount \(\Delta h=\left(V_{0} / A\right)(\beta-3 \alpha) \Delta T\). (b) For a typical system, such as a mercury thermom- eter, why is it a good approximation to neglect the expansion of the flask? expansion of the flask?
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