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Chapter 19: Problem 21

The compression and rarefaction associated with a sound wave propogating in a gas are so much faster than the flow of heat in the gas that they can be treated as adiabatic processes. a) Find the speed of sound, \(v_{s}\), in an ideal gas of molar mass \(M\). b) In accord with Einstein's refinement of Newtonian mechanics, \(v_{\mathrm{s}}\) cannot exceed the speed of light in vacuum, \(c\). This fact implies a maximum temperature for an ideal gas. Find this temperature. c) Evaluate the maximum temperature of part (b) for monatomic hydrogen gas (H). d) What happens to the hydrogen at this maximum temperature?

Short Answer

Expert verified
Answer: The maximum temperature for an ideal gas of monatomic hydrogen is approximately \(2.83 \times 10^{10}\,\text{K}\). At this temperature, hydrogen will transform into a highly ionized plasma.

Step by step solution

01

Write down the ideal gas law and the adiabatic process equation

The ideal gas law is given by \(PV = nRT\), and the adiabatic process equation for an ideal gas is \(PV^{\gamma} = \text{constant}\), where \(P\) is the pressure, \(V\) is the volume, \(n\) is the number of moles, \(R\) is the universal gas constant, \(T\) is the temperature, and \(\gamma\) is the ratio of specific heat capacities.
02

Relate the two equations and find the formula for the speed of sound

Using the ideal gas law, replace \(V\) in the adiabatic process equation with \(\frac{nRT}{P}\). We get \(P\left(\frac{nRT}{P}\right)^{\gamma} = \text{constant}\). Now, differentiate this equation with respect to time and set the result equal to zero, since the process is adiabatic (constant pressure). We get \(\frac{dP}{dT}(\frac{nRT}{P})^{\gamma}=0\). From here, we can determine the speed of sound using the formula \(v_s = \sqrt{\frac{dP}{dT}}\).
03

Calculate the speed of sound in an ideal gas of molar mass \(M\)

We have found the equation for the speed of sound as \(v_s = \sqrt{\frac{dP}{dT}}\). Now we can rewrite it in terms of the molar mass \(M\). Since \(n = \frac{m}{M}\), we have \(v_s = \sqrt{\frac{\gamma RT}{M}}\), where \(m\) is the mass of the gas and \(\gamma\) is the ratio of specific heat capacities. b) In accord with Einstein's refinement of Newtonian mechanics, \(v_{\mathrm{s}}\) cannot exceed the speed of light in vacuum, \(c\). This fact implies a maximum temperature for an ideal gas. Find this temperature.
04

Apply the speed of light limit to find the maximum temperature

According to Einstein, the speed of sound \(v_s\) cannot exceed the speed of light \(c\). So, we set \(v_s \leq c\), which means \(\sqrt{\frac{\gamma RT}{M}} \leq c\). We need to find the maximum temperature, so we solve for \(T\) in the equation. We get \(T \leq \frac{Mc^2}{\gamma R}\). c) Evaluate the maximum temperature of part (b) for monatomic hydrogen gas (H).
05

Calculate the maximum temperature for monatomic hydrogen gas

For monatomic hydrogen gas, we have \(M = 1.0079 \times 10^{-3}\,\text{kg/mol}\) and \(\gamma = \frac{5}{3}\). Now, we can substitute these values into the maximum temperature formula. We get \(T \leq \frac{(1.0079 \times 10^{-3})(3 \times 10^8)^2}{\frac{5}{3} \times 8.314} \approx 2.83 \times 10^{10}\,\text{K}\). d) What happens to the hydrogen at this maximum temperature?
06

Discuss the changes in hydrogen gas at the maximum temperature

At the maximum temperature of about \(2.83 \times 10^{10}\,\text{K}\), the hydrogen atoms would have so much kinetic energy that they would no longer behave like a regular gas. The hydrogen would likely transform from its gaseous state into a highly ionized plasma consisting of free atomic nuclei and electrons. Additionally, nuclear fusion reactions may also occur at this extreme temperature, similar to the processes happening inside stars.

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Most popular questions from this chapter

Which of the following gases has the highest rootmean-square speed? a) nitrogen at \(1 \mathrm{~atm}\) and \(30^{\circ} \mathrm{C}\) b) argon at \(1 \mathrm{~atm}\) and \(30^{\circ} \mathrm{C}\) c) argon at \(2 \mathrm{~atm}\) and \(30^{\circ} \mathrm{C}\) d) oxygen at 2 atm and \(30^{\circ} \mathrm{C}\) e) nitrogen at \(2 \mathrm{~atm}\) and \(15^{\circ} \mathrm{C}\)

Molar specific heat at constant pressure, \(C_{p}\), is larger than molar specific heat at constant volume, \(C_{V}\), for a) a monoatomic ideal gas. b) a diatomic atomic gas. c) all of the above. d) none of the above.

Liquid nitrogen, which is used in many physics research labs, can present a safety hazard if a large quantity evaporates in a confined space. The resulting nitrogen gas reduces the oxygen concentration, creating the risk of asphyxiation. Suppose \(1.00 \mathrm{~L}\) of liquid nitrogen \(\left(\rho=808 \mathrm{~kg} / \mathrm{m}^{3}\right)\) evaporates and comes into equilibrium with the air at \(21.0^{\circ} \mathrm{C}\) and \(101 \mathrm{kPa}\). How much volume will it occupy?

1 .00 mol of molecular nitrogen gas expands in volume very quickly, so no heat is exchanged with the environment during the process. If the volume increases from \(1.00 \mathrm{~L}\) to \(1.50 \mathrm{~L},\) determine the work done on the environment if the gas's temperature dropped from \(22.0^{\circ} \mathrm{C}\) to \(18.0^{\circ} \mathrm{C}\). Assume ideal gas behavior.

What is the total mass of all the oxygen molecules in a cubic meter of air at normal temperature \(\left(25^{\circ} \mathrm{C}\right)\) and pressure \(\left(1.01 \cdot 10^{5} \mathrm{~Pa}\right) ?\) Note that air is about \(21 \%\) (by volume) oxygen (molecular \(\mathrm{O}_{2}\) ), with the remainder being primarily nitrogen (molecular \(\mathrm{N}_{2}\) ).
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