Question

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

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

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.

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 \(v_s = \sqrt{\frac{dP}{dT}}\).

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.

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).

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?

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.

Key concepts

Adiabatic Processes

Adiabatic processes occur in a system where no heat is exchanged with the surroundings. They are crucial in understanding how sound waves travel through a gas. Sound moves through compressions and rarefactions, quickly altering pressure and volume. This rapid change happens faster than heat can be transferred. Due to this speed, these processes are considered adiabatic.

This means that the energy changes in the gas are contained within the system. In adiabatic conditions, the equation used is \(PV^{\gamma} = \text{constant}\), where \(\gamma\) is the heat capacity ratio. This formula helps us relate pressure and volume in an ideal gas while no heat is lost or gained. Understanding adiabatic processes helps in calculating the speed of sound, as they show that sound waves travel without heat exchanging, adhering to specific mathematical relationships.

Maximum Temperature of Gases

In certain extreme conditions, gases can only reach a limited temperature. According to Einstein's theories, sound speed in a gas cannot surpass the speed of light. This relationship sets an upper limit on the speed—and thereby temperature—of the gas.

This maximum temperature can be calculated with the formula \[T \leq \frac{Mc^2}{\gamma R}\], where \(M\) is molar mass, \(c\) is the speed of light, \(R\) is the universal gas constant, and \(\gamma\) is the heat capacity ratio. These limits play a fundamental role in understanding the behaviors of gases under extreme conditions. Calculating the maximum temperature helps predict how the gas reacts as it is heated towards higher energy states, ensuring it remains within the physically possible limits.

Einstein's Mechanics Limitations

Einstein's insight into mechanics revealed that certain physical properties, like the speed of sound, have limits. His relativity theory implies the ultimate speed limit in the universe is that of light, \(c\). That means no object or wave, including sound, can move faster than this speed.

For sound in gases, this means the temperature cannot increase indefinitely. As temperature rises, the kinetic energy and potentially the speed of sound in the gas rise too. However, Einstein's limitation shows that these factors cannot increase beyond the threshold set by the speed of light. This insight offers important boundaries within which scientists must work—vital when exploring high-energy environments such as stellar interiors.

Ideal Gas Law

The ideal gas law is a cornerstone of classical physics, providing a relation between the state variables of gas. The law is expressed as \(PV = nRT\), where \(P\) is pressure, \(V\) is volume, \(n\) is the amount of substance in moles, \(R\) is the ideal gas constant, and \(T\) is temperature.

This equation is fundamental in calculating the speed of sound in an ideal gas, allowing us to connect thermal and dynamic properties. By substituting variables and manipulating this formula, scientists can derive important relationships, like the adiabatic process equation and sound speed formula. The ideal gas law provides the foundation for understanding more complex thermodynamic processes in gases.

Gamma (Heat Capacity Ratio)

The heat capacity ratio, commonly represented by \(\gamma\), is the ratio of specific heat at constant pressure (\(C_p\)) to specific heat at constant volume (\(C_v\)). It plays a pivotal role in characterizing adiabatic processes and is integral to understanding the behavior of gases.

For an ideal gas in adiabatic conditions, \(\gamma\) determines how much the temperature and pressure change with volume. It also appears in the speed of sound formula, \(v_s = \sqrt{\frac{\gamma RT}{M}}\), linking thermal properties with dynamics. Different gases have different \(\gamma\) values: For monatomic gases like hydrogen, \(\gamma\) is 5/3. A solid grasp of \(\gamma\) is essential for working with thermodynamic equations that explain gas behavior.