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Chapter 1: Problem 97

Determine the speed of sound at \(20^{\circ} \mathrm{C}\) in (a) air, (b) helium, and (c) natural gas (methane). Express your answer in \(\mathrm{m} / \mathrm{s}\).

Short Answer

Expert verified
The speed of sound at \(20^{\circ} \mathrm{C}\) in air is approximately \(343 \frac{m}{s}\), in helium is approximately \(1007 \frac{m}{s}\), and in methane is approximately \(446 \frac{m}{s}\).

Step by step solution

01

Convert the Temperature to Kelvin

We are given the temperature in Celsius, but we need it in Kelvin for the formula. The conversion from Celsius to Kelvin is \(K = ^{\circ}C + 273\). So, \(20^{\circ}C = 293K\).
02

Calculate the Speed of Sound in Air

Air mostly consists of nitrogen and oxygen. For this mix, the adiabatic index \(\gamma\) is approximately 1.4 and the specific gas constant \(R\) is about 287 \(\frac{J}{kg \cdot K}\). Substituting these values into the formula, along with the temperature in Kelvin, the speed of sound in air at 20 degrees Celsius is \(v = \sqrt(1.4 \cdot 287 \cdot 293) \approx 343 \frac{m}{s}\)
03

Calculate the Speed of Sound in Helium

For helium, the adiabatic index \(\gamma\) is about 5/3 (or 1.666) and the specific gas constant \(R\) is about 2077 \(\frac{J}{kg \cdot K}\). Using these values in the speed formula, the speed of sound in helium at 20 degrees Celsius becomes \(v = \sqrt(1.666 \cdot 2077 \cdot 293) \approx 1007 \frac{m}{s}\)
04

Calculate the Speed of Sound in Methane

For methane, the adiabatic index \(\gamma\) is about 1.31 and the specific gas constant \(R\) is about 518 \(\frac{J}{kg \cdot K}\). Applying these values in the speed formula, the speed of sound in methane at 20 degrees Celsius is \(v = \sqrt(1.31 \cdot 518 \cdot 293) \approx 446 \frac{m}{s}\)

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Most popular questions from this chapter

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