Problem 39 - An Introduction to Thermal Physics

Question

By applying Newton's laws to the oscillations of a continuous medium, one can show that the speed of a sound wave is given by $$c_{s}=\sqrt{\frac{B}{\rho}}$$ where \(\rho\) is the density of the medium (mass per unit volume) and \(B\) is the bulk modulus, a measure of the medium's stiffness. More precisely, if we imagine applying an increase in pressure \(\Delta P\) to a chunk of the material, and this increase results in a (negative) change in volume \(\Delta V\), then \(B\) is defined as the change in pressure divided by the magnitude of the fractional change in volume: $$B \equiv \frac{\Delta P}{-\Delta V / V}$$ This definition is still ambiguous, however, because I haven't said whether the compression is to take place isothermally or adiabatically (or in some other way). (a) Compute the bulk modulus of an ideal gas, in terms of its pressure \(P,\) for both isothermal and adiabatic compressions. (b) Argue that for purposes of computing the speed of a sound wave, the adiabatic \(B\) is the one we should use. (c) Derive an expression for the speed of sound in an ideal gas, in terms of its temperature and average molecular mass. Compare your result to the formula for the rms speed of the molecules in the gas. Evaluate the speed of sound numerically for air at room temperature. (d) When Scotland's Battlefield Band played in Utah, one musician remarked that the high altitude threw their bagpipes out of tune. Would you expect altitude to affect the speed of sound (and hence the frequencies of the standing waves in the pipes)? If so, in which direction? If not, why not?

Step-by-Step Solution

Answer

(a) Isothermal: \( B = P \), Adiabatic: \( B = \gamma P \); (b) Use adiabatic \( B \); (c) \( c_s = \sqrt{\frac{\gamma RT}{M}} \), about 343 m/s; (d) Altitude affects temperature, potentially lowering sound speed.

Step 1: Compute Bulk Modulus for Isothermal Compression

For an isothermal compression, we use the ideal gas law: \[ PV = nRT \]Assuming a small change, we get: \[ P \Delta V + V \Delta P = 0 \]This gives \( \Delta P = -\frac{P \Delta V}{V} \). Therefore the bulk modulus \( B \) is:\[ B_{\text{isothermal}} = \frac{\Delta P}{-\Delta V / V} = P \].

Step 2: Compute Bulk Modulus for Adiabatic Compression

For adiabatic compression, we use the relation: \[ PV^\gamma = \, \text{constant} \]Differentiating, we get:\[ \gamma PV^{\gamma-1} \Delta V + V^\gamma \Delta P = 0 \]So, \( \Delta P = -\gamma \frac{P \Delta V}{V} \), and the bulk modulus \( B \) is:\[ B_{\text{adiabatic}} = \frac{\Delta P}{-\Delta V / V} = \gamma P \].

Step 3: Justification for Using Adiabatic Compression

The propagation of sound in gases happens rapidly, not allowing enough time for heat exchange with the surroundings. Therefore, the process is adiabatic, making the adiabatic bulk modulus \( B_{\text{adiabatic}} = \gamma P \) the appropriate choice for computing sound speed.

Step 4: Derive the Speed of Sound in an Ideal Gas

Starting with the speed of sound formula: \[ c_{s} = \sqrt{\frac{B}{\rho}} \]and using \( B_{\text{adiabatic}} = \gamma P \), we substitute in: \[ c_{s} = \sqrt{\frac{\gamma P}{\rho}} \]Using \( P = \frac{nRT}{V} \) and \( \rho = \frac{m}{V} \) where \( m \) is the total mass, gives:\[ c_{s} = \sqrt{\frac{\gamma nRT}{m}} = \sqrt{\frac{\gamma RT}{M}} \].This must be compared to the root mean square speed \( v_{\text{rms}} = \sqrt{\frac{3RT}{M}} \).

Step 5: Numerical Evaluation for Air at Room Temperature

Assuming air is primarily diatomic with \( \gamma \approx 1.4 \), and \( T = 298 \, \text{K} \) (room temperature), \( R = 8.314 \, \text{J/mol}\cdot \text{K} \), and \( M \approx 0.029 \, \text{kg/mol} \), the speed of sound is:\[ c_{s} = \sqrt{\frac{1.4 \times 8.314 \times 298}{0.029}} \approx 343 \, \text{m/s} \].

Step 6: Effect of Altitude on Speed of Sound

At high altitudes, air pressure decreases, but at a stable temperature, the speed of sound primarily depends on temperature, remaining relatively constant. However, colder temperatures at altitude can reduce the speed slightly, thus affecting sound and musical synchrony.

Key Concepts

Newton's laws

Newton's laws serve as the foundation for understanding various physics phenomena, including sound waves. Specifically, sound waves are longitudinal waves that propagate through a medium due to oscillations and compressions. Newton's second law, which states that force equals mass times acceleration (\( F = ma \)), is particularly useful in deriving the speed of sound in a medium. Here, this concept helps explain how changes in pressure influence particle motion within the medium, thereby impacting sound propagation.

When considering sound waves, we think about how particles are displaced and returned to their original positions, much like a pendulum oscillates due to gravity. The restoring force that causes particles to oscillate back and forth in a medium is related to the pressure difference that drives the wave. Newton's laws allow us to understand these mechanics more deeply by providing the tools to mathematically describe these forces and resultant accelerations in the medium.- The first law highlights that in the absence of external forces, a body in motion stays in motion, essential for wave propagation.- The third law states that every action has an equal and opposite reaction, underlying the pressure changes seen in oscillating sound waves.

Bulk Modulus

The bulk modulus is a measure of a material's resistance to uniform compression, crucial for calculating sound wave speed. It combines with density to influence sound propagation speeds as seen in the formula \( c_{s} = \sqrt{\frac{B}{\rho}} \).

Imagine squeezing a balloon: the bulk modulus determines how much the balloon resists being squished. It's defined as the ratio of pressure change to fractional volume change:\[B = \frac{\Delta P}{-\Delta V / V}\]Understanding the bulk modulus can be challenging, as it relies on the specific conditions of compression - isothermal or adiabatic.

- **Isothermal Compression:** Occurs at constant temperature and is defined by the ideal gas law:\( PV = nRT \).Here, the bulk modulus corresponds to the pressure change when the volume changes slightly but temperature remains stable. Under these conditions, the equation simplifies to \( B_{\text{isothermal}} = P \).- **Adiabatic Compression:** Involves no heat exchange with the environment. The relationship \( PV^\gamma = \, \text{constant} \)shows the dependency on the adiabatic index \( \gamma \),resulting in \( B_{\text{adiabatic}} = \gamma P \).Adiabatic conditions are significant for sound waves, as sound travels too quickly for heat exchange, making this the more appropriate bulk modulus value.

Isothermal Compression

Isothermal compression refers to a compression process that occurs at a constant temperature. In the context of an ideal gas, this is described by the law \( PV = nRT \),where \( P \)is pressure, \( V \)is volume, \( n \)is the number of moles, \( R \)is the gas constant, and \( T \)is temperature.

During isothermal compression, the ideal gas law shows that any change in volume is directly countered by a change in pressure, such that temperature remains constant. When determining the bulk modulus under these conditions:- The formula simplifies due to the constant temperature, where \( B_{\text{isothermal}} = P \),implying that the stiffness or resistance to compression equals the pressure itself.This form of compression can be seen in systems where the gas is in thermal equilibrium with its surroundings, allowing heat to be exchanged so the temperature does not change. From a practical standpoint, isothermal processes are important in understanding everyday phenomena where temperature remains constant, though this is less typical for sound propagation.

Adiabatic Compression

In adiabatic compression, the system is insulated so that no heat enters or leaves during the process. This means any change in volume results in a significant temperature change, crucial for evaluating sound speed because sound waves propagate using rapid pressure variations without heat exchange.

The principle of adiabatic compression is expressed by \( PV^\gamma = \text{constant} \),where \( \gamma \)is the heat capacity ratio, also known as the adiabatic index, distinct for different gases. This index reflects how much more pressure will increase relative to an equivalent isothermal process.

- Adiabatic compression results in a bulk modulus represented by \( B_{\text{adiabatic}} = \gamma P \).This relates back to the speed of sound in a medium through the equation \( c_{s} = \sqrt{\frac{\gamma P}{\rho}} \),indicating that stiffness and density dictate how fast sound travels.
Adiabatic conditions underscore the mechanics of sound waves since the rapidity of sound results in negligible heat transfer. Thus, these processes and related coefficient are paramount when dealing with sound waves in gases.