Heat and Thermodynamics: Structured Study Notes for PHY 103

Temperature, Heat, and Thermodynamic Laws

Key Concepts in Heat and Thermodynamics

Heat and thermodynamics are foundational topics in physics, describing the behavior of energy transfer, temperature, and the laws governing these processes. Understanding these concepts is essential for analyzing physical systems, phase changes, and energy conservation.

• Temperature: A measure of the average kinetic energy of the particles in a substance.

• Heat: The transfer of energy between objects due to a temperature difference.

• Thermal Contact: When two objects can exchange energy.

• Thermal Equilibrium: When two objects in thermal contact cease to exchange energy.

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

Temperature Scales and Thermometers

Temperature can be measured using various scales and devices, each based on different physical properties that change with temperature.

• Kelvin (Thermodynamic) Scale: Absolute scale; ice point at 273.15 K, triple point at 273.16 K, steam point at 373.15 K.

• Celsius Scale: Ice point at 0°C, steam point at 100°C, divided into 100 equal intervals.

• Fahrenheit Scale: Ice point at 32°F, steam point at 212°F, divided into 180 equal intervals.

• Thermometers: Devices that measure temperature based on thermometric properties such as length/volume of a liquid, resistance, pressure, or color.

Conversion Formulas:

• $T_K = T_C + 273.15$

• $T_F = \frac{9}{5} T_C + 32$

• $T_C = \frac{5}{9}(T_F - 32)$

Defining a Temperature Scale:

• Choose a substance and thermometric property.

• Select two fixed points and measure property values.

• Assign values to the degree of hotness using:

$T_m = \frac{X_m - X_L}{X_U - X_L} N + T_L$

Where $N$ is the fundamental interval, $X_m$ is the property at the point of interest, $X_U$ and $X_L$ are properties at upper and lower fixed points, $T_L$ is the lower fixed point temperature.

Fixed Points and Calibration

Fixed points are used to calibrate thermometers, ensuring agreement at specific temperatures. Common fixed points include the triple point of water, ice point, and steam point.

Thermal Expansion

Expansion of Solids and Liquids

When temperature increases, most solids and liquids expand. The change in dimensions is proportional to the temperature change.

• Linear Expansion: $\Delta L = \alpha L_i \Delta T$

• Volume Expansion: $\Delta V = \gamma V_i \Delta T$

• Area Expansion: $\Delta A = \beta A_i \Delta T$

• For solids: $\gamma = 3\alpha$, $\beta = 2\alpha$

Example: Expansion joints in bridges and railways accommodate thermal expansion.

Heat Capacity and Specific Heat

Definitions and Formulas

Heat capacity ($C$) is the energy required to raise the temperature of a sample by 1°C. Specific heat ($c$) is the energy required to raise the temperature of 1 kg of a substance by 1°C.

• $C = \frac{Q}{\Delta T}$

• $c = \frac{C}{m} = \frac{Q}{m \Delta T}$

• $Q = m c \Delta T$

Latent Heat

Latent heat ($L$) is the energy required for a phase change without temperature change.

• $Q = m L$

• Types: Latent heat of fusion (melting), vaporization (boiling), sublimation (solid to gas).

Ideal Gas Law and Kinetic Theory

Macroscopic Description of an Ideal Gas

An ideal gas follows the equation of state:

• $PV = nRT$

• $PV = N k_B T$

Where $P$ is pressure, $V$ is volume, $n$ is moles, $R$ is gas constant, $N$ is number of molecules, $k_B$ is Boltzmann constant, $T$ is temperature.

Kinetic Theory of Gases

The pressure exerted by a gas is due to collisions of molecules with the container walls. The average kinetic energy per molecule is proportional to temperature.

• $\frac{1}{2} m v^2 = \frac{3}{2} k_B T$

• Degrees of freedom: Monoatomic (3), Diatomic (5), Polyatomic (6)

First Law of Thermodynamics

Energy Conservation in Thermodynamic Systems

The first law states that the change in internal energy ($\Delta E_{int}$) of a system is equal to the heat added ($Q$) minus the work done ($W$):

• $\Delta E_{int} = Q - W$

• Positive $Q$: energy enters system; positive $W$: work done by system.

Special Processes

• Isolated System: No heat or work exchange; $\Delta E_{int} = 0$

• Cyclic Process: Returns to initial state; $\Delta E_{int} = 0$, $Q = W$

• Isobaric Process: Constant pressure; $W = P (V_f - V_i)$

• Isochoric Process: Constant volume; $W = 0$, $\Delta E_{int} = Q$

• Isothermal Process: Constant temperature; $\Delta E_{int} = 0$, $Q = W$

• Adiabatic Process: No heat exchange; $Q = 0$, $\Delta E_{int} = -W$

Second Law of Thermodynamics

Heat Engines and Efficiency

A heat engine converts heat energy into work, operating between a hot and cold reservoir. Efficiency ($e$) is defined as:

• $e = \frac{W}{|Q_h|} = 1 - \frac{|Q_c|}{|Q_h|}$

• Perfect efficiency ($e = 1$) is impossible; some energy is always expelled to the cold reservoir.

Heat Pumps and Refrigerators

Heat pumps and refrigerators transfer heat from cold to hot reservoirs, requiring work input. The coefficient of performance (CoP) is:

• Cooling mode: $CoP = \frac{|Q_c|}{W}$

• Heating mode: $CoP = \frac{|Q_h|}{W}$

Carnot Engine and Maximum Efficiency

The Carnot engine is an ideal reversible engine. Its efficiency is:

• $e = 1 - \frac{T_c}{T_h}$

• Maximum efficiency is achieved only if $T_c = 0$ K, which is unattainable.

Third Law of Thermodynamics

Absolute Zero and Entropy

The third law states that the entropy of a perfect crystal at absolute zero is zero. Absolute zero (0 K) is the lowest possible temperature, where all motion ceases.

• Entropy ($S$): $S = k_B \ln \Omega$

• It is impossible to reach absolute zero by any means (Nernst statement).

Energy Transfer Mechanisms

Conduction

Heat transfer by conduction occurs as molecules in a hotter region transfer energy to adjacent cooler regions. The rate of energy transfer is:

• $P = \frac{Q}{\Delta t} = kA \frac{T_h - T_c}{\Delta x}$

• $P = \frac{T_h - T_c}{R}$, where $R = \frac{\Delta x}{kA}$

Convection

Convection occurs as heated fluid rises due to decreased density, carrying heat through the fluid.

Radiation

All objects radiate energy as electromagnetic waves. The rate of radiation is given by Stefan-Boltzmann law:

• $P_{rad} = \sigma \epsilon A T^4$

• $P_{net} = \sigma \epsilon A (T^4 - T_{env}^4)$

Example: The Sun's surface temperature can be calculated using its power output and radius.

Summary Table: Temperature Scales

Scale	Ice Point	Steam Point	Interval	Conversion
Kelvin	273.15 K	373.15 K	100	$T_K = T_C + 273.15$
Celsius	0°C	100°C	100	$T_C = T_K - 273.15$
Fahrenheit	32°F	212°F	180	$T_F = \frac{9}{5} T_C + 32$

Summary Table: Types of Thermometers

Type	Thermometric Property	Range
Liquid-in-glass	Length/volume of liquid	-35°C to 350°C (mercury)
Gas (constant volume)	Pressure	-270°C to 1500°C
Resistance	Electrical resistance	-200°C to 1000°C
Thermocouple	EMF	-270°C to 2000°C

Summary Table: Energy Transfer Mechanisms

Mechanism	Description	Formula
Conduction	Direct transfer via collisions	$P = kA \frac{T_h - T_c}{\Delta x}$
Convection	Transfer via fluid movement	N/A
Radiation	Transfer via electromagnetic waves	$P_{rad} = \sigma \epsilon A T^4$