Comprehensive Study Notes on Heat, Temperature, and Thermal Expansion

Fundamental Concepts of Heat and Temperature

• Conceptual Inquiries into Heat and Coldness

  • When picking up ice blocks, hands feel cold because thermal energy is transferred from the hands to the ice.

  • Touching a laundry iron base while it is plugged in causes a sensation of extreme heat because thermal energy is transferred rapidly from the iron to the hand.

  • Heat and temperature are fundamental to events ranging from the warmth of the sun to complex systems of thermal equilibrium and heat transfer.

• Biographical Note: Anders Celsius (1701-1744)

  • Anders Celsius was a Swedish scientist active as an astronomer, physicist, and mathematician.

  • He served as a professor of astronomy at Uppsala University from 1730 to 1744.

  • Between 1732 and 1735, he visited observatories in Germany, Italy, and France.

  • He founded the Uppsala Astronomical Observatory in 1741.

  • In 1742, he proposed an inverted form of the Centigrade temperature scale, which was later named the Celsius scale in his honor.

• Thermal Energy

  • Thermal energy is the most basic form of energy contained within a system or an atom.

  • It is not in transit; it remains stored in the system as kinetic energy (KE).

  • It is responsible for the random movements (randomness) of molecules or atoms; increasing thermal energy increases the speed of these particles.

  • Internal Energy: Includes both the kinetic energy and potential energy of molecules within a substance. The kinetic energy component is directly related to the thermal energy of the system.

  • Sources of Thermal Energy: Primary natural sources include the sun and fire. Man-made sources include geothermal energy, nuclear heat, waste heat, electrical energy, chemical energy, and wind energy.

  • Generation through Friction: Rubbing hands together on a cold day transforms the kinetic energy of the hands into thermal energy via friction.

  • Molecular Velocity: When a substance is heated, the average kinetic energy of its molecules increases, leading to more frequent and energetic collisions.

• Heat Energy

  • Heat refers to the transfer of thermal energy between systems as a result of a temperature difference.

  • Heat is energy in transit or an exchange amount of energy.

  • Thermal Equilibrium: This state is reached when two objects in thermal contact attain the same temperature and energy transfer stops.

  • Perception: Heat is not a physical object and cannot be seen; it is sensed through touch.

  • Comparison of Energy Levels: A full bucket of warm water contains more total heat energy than a small red-hot iron nail. However, if the nail is immersed in the water, heat flows from the nail to the water because the nail is at a higher temperature.

  • Molecular Definition: The sum of the kinetic energy of all molecules in a substance is the measure of the heat energy contained in it.

  • Measurement and Units:

    • The SI unit of heat is the Joule ($J$).

    • A common unit is the calorie ($cal$), where $1\,cal = 4.18\,J$.

    • Measurement is performed using a calorimeter.

Comparison between Thermal Energy and Heat

Temperature

• Definition: Temperature is the degree of hotness or coldness of a body. It determines the direction of heat flow.

• Thermal State: When two bodies are in contact, the cold body warms and the hot body cools until they reach the same temperature.

• Macroscopic Parameter: Temperature is defined for a system of a large number of particles, not for a single particle.

• Molecular Concept: Temperature is a measure of the average kinetic energy of the molecules in a body.

• Units:

  • SI System: Kelvin ($K$).

  • Common units: degree Celsius ($^{\circ}C$) and degree Fahrenheit ($^{\circ}F$).

Effects of Heat

• Thermoelectric Effect: If a temperature difference is maintained between two junctions of a closed loop of dissimilar metal wires, an electromotive force (emf) is developed, producing electric current.

• Generation of Light: Substances start emitting light when heated to a sufficiently high temperature.

• Change in Electrical Resistance: Increasing temperature increases the resistance of a conductor but decreases the resistance of a semiconductor.

• Physical Property Changes: Heat application can change temperature, volume, state, color, viscosity, elasticity, solubility rate, rate of chemical reaction, and surface tension.

• Chemical Property Changes: Examples include the ignition of paper or the boiling of an egg.

Thermal Equilibrium and Thermodynamics

• Thermal Equilibrium: Two physical systems are in thermal equilibrium if there is no net flow of thermal energy between them when in thermal contact. At equilibrium, heat exchange still occurs but at the same rate in opposite directions.

• Mercury Thermometer Principle: To measure temperature, the thermometer bulb is brought into thermal contact with an object. After reaching thermal equilibrium, the thermometer registers its own temperature, which by inference is the temperature of the object.

• Zeroth Law of Thermodynamics: Formulated by Ralph Howard Fowler in 1931. It states: "When two bodies A and B are in thermal equilibrium separately with a third body C, then the first two bodies A and B themselves must be in thermal equilibrium."

  • Adiabatic Wall: A non-conducting wall that prevents heat exchange.

  • Diathermic Wall: A conducting wall that allows heat exchange.

Thermometry

• Definition: The science of temperature and its measurement.

• Working Principle: Changes in temperature cause changes in measurable physical properties including:

  1. Length of a bar.

  2. Volume of a liquid.

  3. Volume of a gas at constant pressure.

  4. Pressure of a gas at constant volume.

  5. Magnetic properties of a substance.

  6. Electrical resistance of a wire.

• Key Steps in Thermometry:

  1. Construction: Ensuring uniform variation in properties.

  2. Calibration: Noting two standard fixed points and dividing the interval into equal parts.

  3. Sensitiveness: The ability to read temperature immediately, read small changes, and avoid large heat exchanges with the object.

• Fixed Points:

  • Upper Fixed Point (UFP): The temperature at which pure water boils under standard atmospheric pressure.

  • Lower Fixed Point (LFP): The temperature at which pure ice melts under standard atmospheric pressure.

Temperature Scales and Relationships

• Comparison of Fixed Points:

• Relation Formula:     $\frac{\text{Reading} - LFP}{UFP - LFP} = \text{Constant}$     $\frac{C - 0}{100 - 0} = \frac{F - 32}{212 - 32} = \frac{K - 273}{373 - 273}$     $\frac{C}{100} = \frac{F - 32}{180} = \frac{K - 273}{100}$     $\frac{C}{5} = \frac{F - 32}{9} = \frac{K - 273}{5}$

• Absolute Scale and Absolute Zero:

  • Temperatures in Kelvin are absolute.

  • Absolute Zero: The lowest possible temperature in the universe, defined as $0\,K$, $-273.15^{\circ}C$, or approximately $-459.67^{\circ}F$.

  • At this temperature, molecular motion ceases, kinetic energy is zero, and the velocity of all molecules is zero.

Worked Examples: Temperature Calibration

• Faulty Thermometer Calculation:

  • Fixed points: $5^{\circ}$ and $95^{\circ}$. Reading: $92^{\circ}$.

  • Formula: $\frac{92 - 5}{95 - 5} = \frac{C}{100}$

  • $\frac{87}{90} = \frac{C}{100} \implies 9C = 870 \implies C = 96.67^{\circ}C$.

• Height of Mercury Thread Calculation:

  • Distance between UFP and LFP = $0.800\,m$.

  • Height above LFP = $0.104\,m$.

  • Equation: $0.800\,m = 100^{\circ}C$.

  • Calculation: $\frac{100}{0.800} \times 0.104 = 13.0^{\circ}C$. Total temperature: $13.0^{\circ}C + 0 = 13.0^{\circ}C$.

Thermal Expansion Introduction

• Definition: The change in size (length, area, or volume) of a substance when heated.

• Molecular Mechanism: Heat causes atoms to vibrate more, increasing the average distance between them.

• Water Anomaly: Water contracts between $0^{\circ}C$ and $4^{\circ}C$ but expands after $4^{\circ}C$.

• Biographical Note: Lord Kelvin (1824-1907):

  • William Thomson (Lord Kelvin) was a British physicist and mathematician.

  • He was a Professor at the University of Glasgow.

  • He established the absolute temperature scale and the concept of absolute zero.

  • He patented the galvanometer and perfected transmissions through submarine cables.

Linear Expansion

• Definition: The increase in length of a linear body (rod/wire) due to heat.

• Proportionality:

  1. Change in length ($\Delta l$) is proportional to original length ($l_1$).

  2. Change in length is proportional to change in temperature ($\theta_2 - \theta_1$).

• Linear Expansivity ($\alpha$): The change in length per unit original length per unit change in temperature.

• Formula:     $\alpha = \frac{l_2 - l_1}{l_1(\theta_2 - \theta_1)}$     $l_2 = l_1(1 + \alpha\Delta\theta)$

• Unit: $K^{-1}$.

• Linear Expansivities Table (Common Substances):

  • Invar: $0.05 \times 10^{-5}\,K^{-1}$

  • Steel: $1.2 \times 10^{-5}\,K^{-1}$

  • Copper: $1.7 \times 10^{-5}\,K^{-1}$

  • Aluminum: $2.6 \times 10^{-5}\,K^{-1}$

• Application in Pendulum Clocks:

  • Period: $t = 2\pi\sqrt{\frac{l}{g}}$.

  • Temperature increase increases length ($l$), which increases the period ($t$), causing the clock to lose time.

  • Temperature decrease decreases length, decreasing the period, causing the clock to gain time.

  • Time loss/gain formula: $\text{Loss or gain} = \frac{1}{2}\alpha\Delta\theta \times \text{Duration}$.

Superficial and Cubical Expansion

• Superficial Expansion: The increase in area of a body when heated.

  • Coefficient of Superficial Expansion ($\beta$): Change in area per unit original area per unit change in temperature.

  • Formula: $A_2 = A_1(1 + \beta\Delta\theta)$.

  • Relationship: $\beta = 2\alpha$.

• Cubical Expansion: The increase in volume of a body when heated.

  • Coefficient of Cubical Expansion ($\gamma$): Change in volume per unit initial volume per unit change in temperature.

  • Formula: $V_2 = V_1(1 + \gamma\Delta\theta)$.

  • Relationship: $\gamma = 3\alpha$.

• Ratios of Coefficients: $\alpha : \beta : \gamma = 1 : 2 : 3$.

Thermal Stress and Force

• Force Set-Up: Mechanical force occurs when an object is prevented from expanding or contracting.

• Formula for Force ($F$):     $F = YA\alpha\Delta\theta$     where $Y$ is Young's Modulus, $A$ is cross-sectional area, and $\alpha\Delta\theta$ is thermal strain.

• Thermal Stress: The stress corresponding to the force:     $\text{Thermal Stress} = \frac{F}{A} = Y\alpha\Delta\theta$

Differential Expansion and Bimetallic Thermostats

• Differential Expansion: The difference in expansion between two bodies of different materials heated through the same temperature range.

• Bimetallic Thermostat: Made of two thin strips (e.g., Brass and Steel) welded together.

  • Upon heating, the material with higher $\alpha$ (Brass) expands more, causing the strip to bend into an arc with the brass on the outer (greater) curvature.

  • Used in electric irons, rice cookers, and heating systems as automatic switches.

• Zero Differential Expansion: Condition where the difference in length ($d$) remains constant:     $l_1\alpha = l_1'\alpha'$

Thermal Expansion of Liquids

• Liquids have two coefficients because the vessel containing them also expands.

• Real Expansion ($\Delta V_r$): The actual change in volume of the liquid.

• Apparent Expansion ($\Delta V_a$): The observed change in volume (ignoring vessel expansion).

• Vessel Expansion ($\Delta V_g$): The increase in volume of the container.

• Relation:     $\Delta V_r = \Delta V_a + \Delta V_g$     $\gamma_r = \gamma_a + \gamma_g$     $\gamma_r = \gamma_a + 3\alpha_g$     where $\gamma_r$ is the coefficient of real expansion, $\gamma_a$ is the coefficient of apparent expansion, and $\gamma_g$ is the coefficient of volume expansion of the vessel.

Variation of Density with Temperature

• Since mass remains constant while volume increases with temperature, density must decrease.

• Formula:     $\rho_2 = \frac{\rho_1}{1 + \gamma\Delta\theta}$     where $\rho_1$ and $\rho_2$ are densities at initial and final temperatures, respectively.

Measurement of Absolute (Real) Expansivity of Liquid

• Dulong and Petit's Experiment (Balanced Column Method):

  • Uses a U-shaped tube filled with liquid (e.g., Mercury).

  • Limb AB is cooled in ice water (temperature $\theta_1$, height $h_1$).

  • Limb CD is heated by steam (temperature $\theta_2$, height $h_2$).

  • A wet cloth is used on the connecting portion BC to prevents heat exchange between limbs.

  • At equilibrium, hydrostatic pressures are equal ($P_1 = P_2$):         $\rho_1 g h_1 = \rho_2 g h_2 \implies \frac{h_1}{h_2} = \frac{\rho_2}{\rho_1}$

  • Substituting density variation:         $\gamma = \frac{h_2 - h_1}{h_1(\theta_2 - \theta_1)}$

  • This method allows for the direct calculation of real expansivity ($\gamma$) without considering the solid vessel's expansion.

Exercises and Sample Problems

• Sample Question 1: Can an object be hotter than another if they are at the same temperature? No, temperature defines the thermal state; at the same temperature, there is no net heat flow.

• Sample Question 2: What is the temperature of a vacuum? Vacuum has no molecules, thus the concept of average kinetic energy (temperature) does not apply to it.

• Sample Question 3: Why does heat flow from high to low temperature? It is a natural process to reach thermal equilibrium as described by thermodynamics.

• Sample Question 4: A steel tape is accurate at $20^{\circ}C$. If used at $10^{\circ}C$ to measure $1\,km$, what reading will it show? Since the tape contracts, it will measure the distance as slightly longer. Reading = $1\,km \times [1 + \alpha(20-10)] = 1.00012\,km$.

• Sample Question 5: Calculate density of silver at $100^{\circ}C$ ($\rho_0 = 10310\,kg/m^3$, $\alpha = 1.9 \times 10^{-5}\,^{\circ}C^{-1}$).

  • $\gamma = 3\alpha = 5.7 \times 10^{-5}\,^{\circ}C^{-1}$.

  • $\rho_{100} = \frac{10310}{1 + (5.7 \times 10^{-5})(100)} = 10252\,kg/m^3$.