Thermal Properties of Matter, Calorimetry, and Simple Harmonic Motion

Thermal Properties of Matter

Specific Heat and Heats of Transformation

When a substance absorbs or releases heat, its temperature may change, or it may undergo a phase change. The amount of heat required depends on the material's specific heat and the heat of transformation for phase changes.

• Specific Heat (c): The amount of heat required to raise the temperature of 1 kg of a substance by 1 K (or 1 °C). It is a measure of how resistant a substance is to temperature change.

• Heat of Transformation (L): The amount of heat required to change the phase of 1 kg of a substance (e.g., melting or boiling) without changing its temperature.

• Key Equations:

  • For temperature change: $Q = mc\Delta T$

  • For phase change: $|Q| = mL$

• Example: Heating water from 10°C to 100°C and then boiling it involves both temperature change and phase change.

Energy Flow Diagrams and Bar Charts

Energy flow diagrams help visualize heat transfer processes. The system (e.g., water) is represented by a box, with arrows indicating heat flow. Energy bar charts show thermal energy (Eth) and phase energy (Eph) before and after heat transfer.

• Thermal Energy (Eth): Related to the kinetic energy of particles and proportional to temperature.

• Phase Energy (Eph): Related to the arrangement and bonding of particles; increases with less order (gas > liquid > solid).

Temperature vs. Heat Transfer Graphs

Graphs of temperature versus heat transfer (Q) show regions of temperature change and regions of phase change. During phase change, temperature remains constant while heat is absorbed or released.

Calorimetry and Thermal Equilibrium

Calorimetry Principles

Calorimetry is the quantitative measurement of heat transfer between substances in thermal contact until equilibrium is reached. The heat lost by one substance equals the heat gained by another (energy conservation).

• Conservation of Energy: $Q_{lost} + Q_{gained} = 0$

• Example: A hot metal dropped into cooler water; both reach the same final temperature.

Calorimetry with Phase Change

When a phase change occurs (e.g., ice melting in water), multiple heat transfer processes must be considered: warming the ice, melting it, and warming the resulting water.

• Heat Transfer Steps:

  1. Heating ice to 0°C: $Q_1 = mc_{ice}\Delta T$

  2. Melting ice: $Q_2 = mL_{fusion}$

  3. Heating melted water: $Q_3 = mc_{water}\Delta T$

Thermal Expansion

Linear and Volumetric Expansion

When a substance changes temperature, its dimensions change due to thermal expansion. The change depends on the material's expansion coefficients.

• Linear Expansion: $\Delta L = \alpha L_{initial} \Delta T$

• Volumetric Expansion: $\Delta V = \beta V_{initial} \Delta T$

• Coefficients: $\alpha$ and $\beta$ (units: K-1)

Applications: Bi-metallic Strips and Galileo Thermometers

Bi-metallic strips bend when heated due to different expansion rates of metals. Galileo thermometers use floating bulbs to indicate temperature based on density changes from thermal expansion.

Heat Transfer Mechanisms

Conduction

Conduction is the transfer of thermal energy through a material. The rate depends on the material's thermal conductivity (k), area (A), thickness (L), and temperature difference ($\Delta T$).

• Equation: $\frac{Q}{\Delta t} = k \frac{A}{L} \Delta T$

• Materials with high k (e.g., metals) conduct heat faster than those with low k (e.g., wood, Styrofoam).

Convection

Convection is the transfer of heat by the movement of fluids (liquids or gases). It occurs in open systems and is influenced by temperature differences and fluid motion.

Radiation

Radiation is the transfer of energy by electromagnetic waves, such as infrared radiation. All objects emit radiation depending on their temperature and emissivity.

• Stefan-Boltzmann Law: $P = e \sigma A T^4$

• $e$ = emissivity, $\sigma$ = Stefan-Boltzmann constant ($5.67 \times 10^{-8}$ W/m2K4), $A$ = area, $T$ = temperature in K

Simple Harmonic Motion (SHM)

Characteristics of SHM

Simple harmonic motion describes periodic oscillations, such as a mass on a spring or a pendulum. The motion is characterized by amplitude (A), period (T), and frequency (f).

• Position as a function of time: $x(t) = A \cos \left( \frac{2\pi t}{T} \right)$

• Frequency: $f = \frac{1}{T}$

• Velocity: $v(t) = -A \frac{2\pi}{T} \sin \left( \frac{2\pi t}{T} \right)$

Mass-Spring System and Hooke's Law

The restoring force in a mass-spring system is described by Hooke's Law:

• Hooke's Law: $F = -kx$

• Period of oscillation: $T = 2\pi \sqrt{\frac{m}{k}}$

• Stiffer springs (higher k) result in shorter periods.

Energy in SHM

Energy in SHM is conserved and alternates between kinetic energy (KE) and potential energy (PE) stored in the spring.

• Kinetic Energy: $KE = \frac{1}{2}mv^2$

• Spring Potential Energy: $PE = \frac{1}{2}kx^2$

• Total Energy: $E_{total} = KE + PE$ (constant)

Waves and Superposition

Wave Pulses and Reflection

Waves are disturbances that travel through a medium. They can be longitudinal (disturbance parallel to direction) or transverse (disturbance perpendicular).

• Reflection at Fixed End: Pulse inverts upon reflection.

• Reflection at Free End: Pulse reflects without inversion.

Tables

Table: Selected Specific Heats, Melting and Boiling Points, and Heats of Transformation

Substance	c (solid) (J/kg·K)	c (liquid) (J/kg·K)	Tmelt (°C)	Lf (J/kg)	Tboil (°C)	Lv (J/kg)
Water	2090	4190	0	3.33 × 105	100	22.6 × 105
Ethyl alcohol	-	2400	-114	1.09 × 105	78	8.79 × 105
Mercury	-	140	-39	0.11 × 105	357	2.96 × 105
Lead	128	-	328	0.25 × 105	1750	8.58 × 105
Aluminum	900	-	-	-	-	-
Copper	385	-	-	-	-	-

Table: Thermal Expansion Coefficients

Substance	Linear: α (K-1)	Volume: β (K-1)
Water	-	210 × 10-6
Alcohol	-	1100 × 10-6
Kerosene	-	990 × 10-6
Glass	9 × 10-6	27 × 10-6
Steel	12 × 10-6	36 × 10-6
Concrete	12 × 10-6	36 × 10-6
Brass	19 × 10-6	57 × 10-6

Table: Thermal Conductivity Values

Material	k (W/m·K)
Copper	400
Aluminum	205
Stainless steel	14
Ice	1.7
Glass	1.05
Skin	0.50
Wood	0.2
Paper	0.05
Styrofoam	0.033
Air	0.026

Table: Emissivity Values

Substance/Material	Emissivity, e
Skin	0.95 – 0.98
Wood	0.9
Polished aluminum	0.05
Brick	0.8 – 0.9
Styrofoam	0.6

Additional info: These notes cover the core concepts of thermal properties of matter, calorimetry, thermal expansion, heat transfer mechanisms, and simple harmonic motion, with relevant equations, diagrams, and tables for college-level physics.