5.2 Heat, Thermodynamics, Waves & Sound

Heat, Temperature Scales, Specific Heat & Latent Heat

Thermodynamics is the branch of physical science concerned with heat, temperature, and their relations to energy, work, and radiation. Understanding heat transfer mechanisms and thermal properties of matter is essential for physics applications tested on the PAF Airman exam.

Heat vs. Temperature

• Heat ($Q$): Thermal energy transferred between systems due to a temperature difference ($\text{Joules}$). Heat always flows spontaneously from a region of higher temperature to a region of lower temperature.
• Temperature ($T$): A measure of the average kinetic energy of the individual molecules within a substance. Temperature determines the direction of heat transfer.

Temperature Scales & Conversions

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Modes of Heat Transfer

Heat transfers through matter and space via three distinct physical mechanisms:

1. Conduction: Direct transfer of thermal energy through molecular collisions without bulk motion of the material. Conduction occurs primarily in solids; metals are excellent thermal conductors due to free electrons.
2. Convection: Heat transfer by the physical bulk movement of fluid (liquid or gas) currents driven by density differences (warm fluid expands, becomes less dense, and rises).
3. Radiation: Transfer of thermal energy via electromagnetic waves (primarily infrared radiation) without requiring any material medium. Solar radiation reaching Earth through the vacuum of space is a primary example.

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Specific Heat Capacity & Latent Heat

Specific Heat Capacity ($c$)

The amount of thermal energy required to raise the temperature of $1\text{ kg}$ of a substance by $1^\circ\text{C}$ (or $1\text{ K}$): $Q = m c \Delta T$ Where $\Delta T = T_{\text{final}} - T_{\text{initial}}$.

Latent Heat ($L$)

Thermal energy absorbed or released during a phase change (solid to liquid, liquid to gas) without any change in temperature: $Q = m L$

• Latent Heat of Fusion ($L_f$): Energy required for melting/freezing ($L_{f,\text{water}} \approx 3.36 \times 10^5\text{ J/kg}$).
• Latent Heat of Vaporization ($L_v$): Energy required for boiling/condensation ($L_{v,\text{water}} \approx 2.26 \times 10^6\text{ J/kg}$).

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Worked Example: Heat Calculation

Problem: An iron block of mass $4\text{ kg}$ at $100^\circ\text{C}$ is immersed into $2\text{ kg}$ of liquid water at $20^\circ\text{C}$. Assuming no heat escapes to the surroundings, find the final equilibrium temperature ($T_f$). (Take $c_{\text{iron}} = 450\text{ J/(kg}\cdot^\circ\text{C)}$ and $c_{\text{water}} = 4200\text{ J/(kg}\cdot^\circ\text{C)}$).

Solution: By the Law of Conservation of Energy: Heat lost by iron = Heat gained by water. $m_{\text{iron}} c_{\text{iron}} (100 - T_f) = m_{\text{water}} c_{\text{water}} (T_f - 20)$ $(4)(450)(100 - T_f) = (2)(4200)(T_f - 20)$ $1800 (100 - T_f) = 8400 (T_f - 20)$ Divide both sides by 600: $3(100 - T_f) = 14(T_f - 20)$ $300 - 3 T_f = 14 T_f - 280$ $580 = 17 T_f \implies T_f = \frac{580}{17} \approx 34.1^\circ\text{C}$

Wave Motion, Sound Properties, Acoustics & Doppler Effect

Waves represent disturbance mechanisms that transport energy and momentum through space or matter without transporting bulk matter.

Classification of Waves

Waves are broadly categorized into mechanical waves (requiring an elastic medium, e.g., sound, water, seismic waves) and electromagnetic waves (propagating through vacuum, e.g., light, radio, X-rays).

Transverse vs. Longitudinal Waves

• Transverse Waves: Particle displacement is perpendicular to the direction of wave propagation (e.g., light waves, plucked guitar strings). Features crests (high points) and troughs (low points).
• Longitudinal Waves: Particle displacement is parallel to the direction of wave propagation (e.g., sound waves in air). Features compressions (high-pressure regions) and rarefactions (low-pressure regions).

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Fundamental Wave Characteristics & Equation

1. Wavelength ($\lambda$): Distance between consecutive identical points in phase (e.g., crest to crest), measured in meters ($\text{m}$).
2. Frequency ($f$): Number of complete wave cycles passing a fixed point per second, measured in Hertz ($\text{Hz} = \text{s}^{-1}$).
3. Period ($T$): Time taken to complete one full cycle ($T = 1 / f$).
4. Wave Velocity ($v$): The propagation speed of the wave front: $v = f \lambda$

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Sound Waves & Acoustic Phenomena

Sound is a longitudinal mechanical wave caused by vibrating sources.

Key Characteristics of Sound

• Loudness: Depends on the amplitude of the wave and human ear sensitivity. Measured on the decibel ($\text{dB}$) sound level scale.
• Pitch: Determined directly by the wave frequency. High frequency produces high pitch (treble); low frequency produces low pitch (bass).
• Quality (Timbre): Depends on the waveform and harmonic overtones, allowing distinction between different instruments playing the same fundamental frequency.

Speed of Sound

The speed of sound depends on the medium's elasticity and density. Sound travels fastest in solids, slower in liquids, and slowest in gases: $v_{\text{solid}} > v_{\text{liquid}} > v_{\text{gas}}$ In dry air at $0^\circ\text{C}$, speed of sound is $\approx 331\text{ m/s}$. At temperature $T$ ($^\circ\text{C}$): $v_{\text{sound}} \approx 331 + 0.6 \, T(^\circ\text{C})$ At room temperature ($20^\circ\text{C}$), $v_{\text{sound}} \approx 343\text{ m/s}$.

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Doppler Effect and Resonance

• Resonance: Occurs when a periodic driving force matches the natural frequency of an oscillating system, resulting in large-amplitude vibration (e.g., tuning forks, musical instruments, structural bridges).
• Doppler Effect: The apparent change in frequency of a wave observed when there is relative motion between the wave source and the observer:
  • When source and observer move toward each other, observed pitch increases ($\uparrow f$).
  • When source and observer move away from each other, observed pitch decreases ($\downarrow f$).

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Worked Example: Wave Velocity & Sound Propagation

Problem: A radar unit emits an ultrasonic sound wave of frequency $40\text{ kHz}$ ($40,000\text{ Hz}$) toward an underwater submarine. If the speed of sound in seawater is $1500\text{ m/s}$:

1. Calculate the wavelength of the ultrasonic wave in seawater.
2. If the echo returns to the radar receiver after $1.2\text{ seconds}$, calculate the distance to the submarine.

Solution:

1. Using the wave equation $v = f \lambda$: $\lambda = \frac{v}{f} = \frac{1500\text{ m/s}}{40,000\text{ Hz}} = 0.0375\text{ m} = 3.75\text{ cm}$

2. Total travel time for the round-trip echo is $t = 1.2\text{ s}$. One-way time to submarine is $t_{\text{one-way}} = 0.6\text{ s}$. $\text{Distance } d = v \cdot t_{\text{one-way}} = (1500\text{ m/s})(0.6\text{ s}) = 900\text{ meters}$