Question

The compression and rarefaction associated with a sound wave propagating 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, \(v_{\mathrm{s}}\), 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 \(\operatorname{gas}(\mathrm{H})\) d) What happens to the hydrogen at this maximum temperature?

Short answer

Answer: At the maximum temperature of \(T_{max} \approx 5.49 \times 10^{12}\ K\), hydrogen atoms would have so much energy that their normal atomic structure would break down. Hydrogen would behave more like a plasma than a gas, and electrons would be stripped from their atomic nuclei. This state is known as the ionization of hydrogen atoms, which is the process by which hydrogen atoms lose their electrons to form ions. In summary, hydrogen at this maximum temperature gets ionized into a plasma state.

Step-by-step solution

a) Find the speed of sound in an ideal gas.

First, we need to find an expression for the speed of sound, \(v_s\), in terms of the ideal gas variables. The speed of sound in an ideal gas can be given by the following equation: $$ v_{s} = \sqrt{\frac{\gamma P}{\rho}} $$ Here, \(\gamma\) is the adiabatic index, \(P\) is the pressure, and \(\rho\) is the density of the gas. Next, we will use the ideal gas law to express the density (\(\rho\)) and pressure (\(P\)) in terms of the molar mass (\(M\)), temperature (\(T\)), and the gas constant (\(R\)): $$ PV = nRT \Rightarrow P = \frac{nRT}{V} $$ And density can be expressed as: $$ \rho = \frac{m}{V} = \frac{nM}{V} $$ Now, we can substitute these equations into the formula for the speed of sound: $$ v_{s} = \sqrt{\frac{\gamma nRT}{nMV}} = \sqrt{\frac{\gamma RT}{M}} $$ So, the speed of sound in an ideal gas of molar mass \(M\) is given by the equation: $$ v_{s} = \sqrt{\frac{\gamma RT}{M}} $$

b) Find the maximum temperature for an ideal gas.

We are given that the speed of sound cannot exceed the speed of light in vacuum, \(c\). Thus, we have the following inequality: $$ v_{s} \leq c $$ Substituting the expression for the speed of sound, we get: $$ \sqrt{\frac{\gamma RT}{M}} \leq c $$ Now, we need to find the maximum temperature, \(T_{max}\), for an ideal gas by solving this inequality for \(T\): $$ T_{max} \leq \frac{c^2 M}{\gamma R} $$

c) Evaluate the maximum temperature for monatomic hydrogen gas (H).

To find the maximum temperature for monatomic hydrogen gas, we will use the maximum temperature equation and substitute the appropriate values for \(\gamma\), \(R\), and \(M\): For monatomic gases, \(\gamma = \frac{5}{3}\), for hydrogen, \(M = 1.0 \times 10^{-3}\ kg/mol\), and the gas constant \(R = 8.314\ J/(mol \cdot K)\): $$ T_{max} \leq \frac{(3.0 \times 10^8 m/s)^2 (1.0 \times 10^{-3} kg/mol)}{\frac{5}{3} (8.314 J/(mol \cdot K))} $$ After calculating the above expression, we get: $$ T_{max} \leq 5.49 \times 10^{12} K $$

d) What happens to the hydrogen at this maximum temperature?

At the maximum temperature of \(T_{max} \approx 5.49 \times 10^{12}\ K\), hydrogen atoms would have so much energy that their normal atomic structure would break down. This would cause hydrogen to behave more like a plasma than a gas, and electrons would be stripped from their atomic nuclei. This state is known as the ionization of hydrogen atoms, which is the process by which hydrogen atoms lose their electrons to form ions. In summary, hydrogen at this maximum temperature gets ionized into a plasma state.

Key concepts

Adiabatic Process

An adiabatic process is a thermodynamic process in which no heat is transferred into or out of the system. In other words, all changes in internal energy are due to work done by or on the system. This is a crucial concept when studying the speed of sound in gases. During the propagation of a sound wave, the gas undergoes rapid compressions and rarefactions which happen so quickly that there is no time for heat to enter or leave the system. Hence, the process is adiabatic, and no temperature change occurs during the sound wave propagation.

Since adiabatic processes do not exchange heat with their surroundings, they are characterized by the adiabatic index \( \gamma \), which is the ratio of specific heats at constant pressure (\( C_p \)) to that at constant volume (\( C_v \)). The value of \( \gamma \) is key in determining the speed of sound in an ideal gas, as it appears in the formula \( v_s = \sqrt{\frac{\gamma P}{\rho}} \), where \( P \) is pressure, and \( \rho \) is density.

Ideal Gas Law

The ideal gas law is a fundamental equation in thermal physics that describes the relationships between pressure (\(P\)), volume (\(V\)), temperature (\(T\)), and the number of moles (\(n\)) of an ideal gas. The equation is given as \(PV = nRT\), where \(R\) is the universal gas constant. The law assumes that the gas particles move randomly, and there are no intermolecular forces affecting them, except during collisions.

To understand the speed of sound in an ideal gas, we use the ideal gas law to relate pressure and density to temperature and molar mass. As seen in the textbook solution, this allows us to express the speed of sound as \(v_{s} = \sqrt{\frac{\gamma RT}{M}}\). By knowing the molar mass (\(M\)) and using the known value of \(R\), and by determining \(T\) and \(\gamma\), we can calculate the speed of sound in that gas. The ideal gas law, therefore, serves as an essential bridge linking macroscopic properties to the speed of sound—an intrinsic property of the gas.

Thermal Physics

Thermal physics involves studying the energy changes within a system, particularly those changes resulting from temperature and heat flow. One of the interesting outcomes in thermal physics is the upper limit of temperature for an ideal gas related to the speed of sound. This is because, according to Einstein’s theory, the speed of sound cannot exceed the speed of light.

In this context, the speed of sound is rooted in the kinetic theory, which postulates that the temperature of a gas is proportional to the average kinetic energy of its particles. Higher temperatures imply higher particle speeds. However, when the speed of sound approaches the speed of light, we reach a temperature limit beyond which the traditional gas laws break down, and the gas could convert into a plasma, as discussed for the hypothetical case of monatomic hydrogen reaching extremely high temperatures in the exercise. Thermal physics provides the framework to understand phenomena like these, bridging the gap between thermal properties and relativistic constraints.