Lecture 26: Temperature, Gas Laws, and Applications of Sound

Temperature and Gas Laws

Overview

This lecture covers the fundamental concepts of temperature, temperature scales, the atomic theory of matter, thermal equilibrium, and applications of sound including the Doppler effect and ultrasound. These topics are essential for understanding the kinetic theory of gases and the behavior of matter at the molecular level.

Applications of Sound

Doppler Effect

The Doppler Effect describes the change in frequency or wavelength of a wave in relation to an observer moving relative to the source of the wave. It is commonly observed with sound waves.

• Stationary Source: The time between wave crests is $T = \frac{\lambda}{v_{sound}}$.

• Moving Source: The observed wavelength changes: $\lambda' = \lambda - v_{source} T$.

• Observed Frequency: For a stationary observer and moving source: $f' = \frac{f}{1 - \frac{v_{source}}{v_{sound}}}$

• Stationary Source, Moving Observer: $f' = f \left(1 + \frac{v_{observer}}{v_{sound}}\right)$

• Both Moving: $f' = f \frac{v_{sound} + v_{observer}}{v_{sound} - v_{source}}$

• Applications: Used in radar, medical imaging (ultrasound), and astronomy.

Example: A train approaches a platform at 10.0 m/s, blowing its whistle. The observer hears a frequency shift due to the Doppler effect. If $v_{sound} = 331$ m/s and $f = 240$ Hz, the observed frequency is $f' = 240 \left(1 - \frac{10}{331}\right) = 232.8$ Hz.

Shock Waves and Sonic Boom

• Shock Waves: Occur when an object moves faster than the speed of sound, compressing air and creating a sharp change in pressure.

• Sonic Boom: The sound associated with shock waves produced by an object traveling at supersonic speeds.

• Examples: Jet aircraft breaking the sound barrier, bullets, and certain animals (e.g., snapping shrimp).

Ultrasound (Sonar & Imaging)

• Ultrasound: High-frequency sound waves (typically 1–15 MHz) used for imaging and detection.

• Applications: Medical imaging, echolocation in bats, and sonar in submarines.

• Wavelength: $\lambda = 0.1 - 1.5$ mm, depending on frequency and medium.

Kinetic Theory of Gases

Atomic Theory of Matter

The atomic theory states that all matter is composed of atoms, which combine to form molecules. Atoms consist of a nucleus (protons and neutrons) and electrons.

• Atomic Mass Unit (u): $1\,u = 1.6605 \times 10^{-27}$ kg

• Carbon-12 Standard: $^{12}C$ has 6 protons and 6 neutrons, mass = 12.000 u

• Brownian Motion: Random movement of particles suspended in a fluid, evidence for atomic theory.

Temperature

Definition and Measurement

Temperature is a measure of how hot or cold an object is, related to the average kinetic energy of its particles.

• Physical Properties: Density, conductivity, color can change with temperature.

• Measurement: Thermometers use physical changes (e.g., expansion of liquid) to measure temperature.

Temperature Scales

There are three main temperature scales: Fahrenheit, Celsius, and Kelvin.

• Fahrenheit (°F): Used in the United States. Water freezes at 32°F and boils at 212°F.

• Celsius (°C): Used internationally. Water freezes at 0°C and boils at 100°C.

• Kelvin (K): The SI unit for temperature, based on absolute zero. Water freezes at 273 K and boils at 373 K.

Conversion Equations:

• $T[°F] = \frac{9}{5}T[°C] + 32$

• $T[°C] = \frac{5}{9}(T[°F] - 32)$

• $T[K] = T[°C] + 273.15$

Temperature Scale Comparison Table

°F	°C	K	Description
212	100	373	Boiling point of water
32	0	273	Freezing point of water
-459	-273	0	Absolute zero
-58	-50	223	Boiling point of nitrogen
-109	-78	195	Freezing point of dry ice (CO2)

Example Table: Temperature Conversion

Object	Fahrenheit	Celsius	Kelvin
Human Body	98.6°F	37°C	310 K
Lowest temperature ever recorded on Earth	-117°F	-82.9°C	190 K
Temperature of Space	-454°F	-270°C	3 K
Surface of the Sun	10540°F	5827°C	6000 K
Tungsten filament inside incandescent lightbulbs	4580°F	2527°C	2800 K

Thermal Equilibrium

Definition and Zeroth Law of Thermodynamics

Thermal equilibrium occurs when two systems in contact have the same temperature and no net heat flow between them.

• Zeroth Law of Thermodynamics: If two systems are each in thermal equilibrium with a third system, then they are in thermal equilibrium with each other.

• Example: A cup of hot coffee and a cup of cold tea placed in a room will eventually reach the same temperature as the room.

Additional info:

• Some images referenced applications of sound (e.g., shock waves, ultrasound in bats, and medical imaging), which are relevant to the study of waves and thermodynamics in physics.

• Atomic theory and temperature scales are foundational for understanding kinetic theory and gas laws, which are covered in college physics courses.