The Timpson Lyceum · University Physics I (2025)

Temperature and Heat - Lecture 17 (University Physics I)

Overview

This lecture opens the thermodynamics unit by building the macroscopic concepts of temperature and heat from the ground up. We start with the operational idea of temperature (something measurable, e.g. by the length of an expanding rod), formalize it with the zeroth law of thermodynamics and the standard temperature scales (Fahrenheit, Celsius, Kelvin, and absolute zero). We then study how matter responds to temperature changes through thermal expansion (linear and volumetric) and the thermal stress that arises when expansion is prevented. Finally we define heat as energy in transit, quantify it with specific heat and latent heat for phase changes, and survey the three modes of heat transfer: conduction, convection, and radiation. These ideas are foundational for the entire thermodynamics sequence.

Key Concepts

Temperature and the zeroth law

• Temperature is a measurable quantity that reflects how much the microscopic particles of a substance are jiggling (a rough measure of internal kinetic energy). It is its own fundamental unit, not built from kilograms, meters, and seconds.
• Heat is not temperature: heat is the energy that transfers between systems because of a temperature difference. "Hot" (high temperature) and "heat" (energy transfer) are distinct ideas.
• Thermal equilibrium: two systems are in thermal equilibrium when no net heat flows between them.
• Zeroth law of thermodynamics: if two systems are each in thermal equilibrium with a third system, they are in thermal equilibrium with each other. This implicitly defines temperature — two systems are in thermal equilibrium if and only if they have the same temperature, even if never placed in contact.
• A good conductor (metal) feels colder than a poor conductor (wood) at the same temperature, because it pulls heat from your hand faster. Feel is about heat flow, not temperature.

Temperature scales

• Fahrenheit and Celsius are both anchored to the freezing point (32 °F = 0 °C) and boiling point (212 °F = 100 °C) of water at one atmosphere. There are 180 Fahrenheit degrees but only 100 Celsius degrees over that same interval, giving the factor $\tfrac{180}{100}=\tfrac{9}{5}$.
• Conversions: $T_F = \tfrac{9}{5}T_C + 32$ and $T_C = \tfrac{5}{9}(T_F - 32)$.
• Absolute zero ($-273.15\,{}^\circ\text{C}$) is where the pressure of an ideal gas extrapolates to zero — intuitively, where particle motion ceases. A real gas can only approach it.
• Kelvin keeps the Celsius degree size but places zero at absolute zero: $T_K = T_C + 273.15$. No degree symbol is used. Water freezes at 273.15 K and boils at 373.15 K. Kelvin is a fundamental SI unit.
• Because the increments match, a temperature change in kelvin equals the same change in Celsius. Use Kelvin whenever an absolute temperature is required.

Thermal expansion

• Linear expansion: the change in length is proportional to both the original length and the temperature change. $\Delta L = \alpha L_0\,\Delta T$ Here $\alpha$ is the coefficient of linear expansion, a material property (e.g. steel $\approx 1.2\times10^{-5}\,\text{K}^{-1}$).
• Volume expansion: $\Delta V = \beta V_0\,\Delta T$, where $\beta$ is the coefficient of volume expansion. For an isotropic solid, $\beta = 3\alpha$ (derived from $V = L^3$, so $dV = 3L^2\,dL$). For liquids and gases only $\beta$ is meaningful.
• A hole in a heated plate gets larger, not smaller — every point of the material scales up as if the removed piece were still there.
• Water is anomalous: it is densest near $4\,{}^\circ\text{C}$ and expands as it cools below that. Ice is about 10% less dense than water, so ice floats and lakes freeze from the top down (the ice layer insulating the water beneath).
• Bimetallic strips: bonding a high-$\alpha$ metal to a low-$\alpha$ metal makes the strip curl when heated, used in dial thermometers and thermostat switches.

Thermal stress

• If a rod is clamped between rigid walls, the thermal expansion it "wants" must be canceled by an elastic compression, so $\Delta L_{\text{thermal}} + \Delta L_{\text{stress}} = 0$. Combining with Young's modulus $Y$ gives the stress required to hold the length fixed: $\frac{F}{A} = -Y\,\alpha\,\Delta T$ Heating produces compression; cooling produces tension. Even modest temperature changes can produce enormous forces.

Quantity of heat, specific heat, and moles

• Calorie: energy to raise 1 g of water by 1 °C; $1\ \text{cal} = 4.186\ \text{J}$. Food "Calories" are kilocalories ($1\ \text{Cal} = 1000\ \text{cal}$).
• Specific heat $c$ measures how much heat a material absorbs per unit mass per degree: $Q = mc\,\Delta T$ Water has a very high specific heat ($c \approx 4186\ \text{J/(kg}\cdot\text{K)}$) versus copper ($\approx 390\ \text{J/(kg}\cdot\text{K)}$), making it a good "thermal inertia" and coolant.
• A mole is a fixed count of particles ($N_A \approx 6.022\times10^{23}$). One mole of a substance has a mass in grams equal to its molar mass $M$ (e.g. H ≈ 1 g/mol, He ≈ 4 g/mol; O₂ ≈ 32 g/mol since it is diatomic). With $m = nM$, the molar heat capacity is $C = Mc$, so $Q = nC\,\Delta T$.

Phase changes and latent heat

• During a phase change the temperature stays constant while heat is added or removed — ice at 0 °C melts before warming further; water at 100 °C boils before its temperature rises.
• Latent heat of fusion $L_f$ (melt/freeze) and latent heat of vaporization $L_v$ (boil/condense): $Q = mL$. For water $L_f = 3.34\times10^5\ \text{J/kg}$ and $L_v \approx 2.256\times10^6\ \text{J/kg}$ (vaporization is far larger — why sweating cools so effectively).
• To carry a substance across several states, sum the heat for each segment separately ($mc\,\Delta T$ within a phase, $mL$ at each transition); note ice ($c \approx 2100$), water ($c \approx 4186$), and steam each have different specific heats.

Heat transfer

• Conduction: heat flows through a material as particles jostle neighbors. The heat current is $H = \frac{dQ}{dt} = kA\,\frac{T_H - T_C}{L}$ with $k$ the thermal conductivity. Larger cross-sectional area $A$ and bigger $\Delta T$ help flow; greater length $L$ hinders it. Thermal resistance $R = L/k$ (high $R$ = good insulator); $k$ and $R$ are inversely related.
• Convection: heat moves by physically transporting the heated mass (a fan, blowing on soup, Earth's mantle). No simple equation.
• Radiation: all bodies emit electromagnetic waves (mostly infrared near room temperature, which is why infrared is loosely called "heat"). The radiated power is the Stefan–Boltzmann law: $H = A e \sigma T^4$ with $e$ the emissivity (surface/material dependent), $\sigma$ the Stefan–Boltzmann constant, and $T$ the absolute temperature in kelvin. A body both emits and absorbs, so the net exchange with surroundings at $T_s$ is $H_{\text{net}} = A e \sigma\,(T^4 - T_s^4)$.

Worked Example

A steel surveyor's tape reads short when warm. A steel tape is calibrated to read exactly correct at $20\,{}^\circ\text{C}$. On a $35\,{}^\circ\text{C}$ day a surveyor reads a distance of $35.794\ \text{m}$ off the tape. Because the tape itself has expanded, each marked interval now spans a slightly greater true length, so the actual distance is longer than the reading. With $\alpha_{\text{steel}} = 1.2\times10^{-5}\ \text{K}^{-1}$ and $\Delta T = 15\,\text{K}$:

The true length is the reading scaled up by the tape's fractional expansion,

A thermal-stress force. An aluminum spacer ($\alpha = 2.4\times10^{-5}\ \text{K}^{-1}$, $Y = 7.0\times10^{10}\ \text{Pa}$, area $A = 20\ \text{cm}^2 = 0.0020\ \text{m}^2$) just fits between two rigid steel walls at $17.2\,{}^\circ\text{C}$. Find the stress and the force on each wall when it warms to $22.3\,{}^\circ\text{C}$ ($\Delta T = 5.1\ \text{K}$):

A mere 5 °C rise generates about 17,000 N — in practice the spacer simply pushes slightly into the walls.

Summary

• Temperature is a measurable, fundamental quantity; heat is energy transferred because of a temperature difference. The zeroth law guarantees that "same temperature" and "thermal equilibrium" mean the same thing.
• Convert scales with $T_F = \tfrac{9}{5}T_C + 32$ and $T_K = T_C + 273.15$; absolute zero ($-273.15\,{}^\circ\text{C}$) is the floor of the Kelvin scale. Use Kelvin for any absolute temperature.
• Solids expand by $\Delta L = \alpha L_0\,\Delta T$ and $\Delta V = \beta V_0\,\Delta T$ with $\beta = 3\alpha$; clamping expansion creates thermal stress $F/A = -Y\alpha\,\Delta T$. Water's density anomaly near 4 °C lets ice float and lakes freeze top-down.
• Heating within a phase follows $Q = mc\,\Delta T$ (water's high $c$ stores lots of energy); crossing a phase boundary follows $Q = mL$ at constant temperature, and vaporization costs far more than fusion.
• Heat travels by conduction ($H = kA\,\Delta T/L$, resistance $R = L/k$), convection (bulk transport of mass), and radiation ($H = Ae\sigma T^4$, with net exchange depending on $T^4 - T_s^4$); near room temperature radiation is mostly infrared.