The study of thermal properties begins with distinguishing between the concepts of temperature and heat . While temperature is a macroscopic measure of the relative hotness or coldness of a system, heat represents the energy transferred across a boundary due to a temperature gradient.
When two bodies of different temperatures are brought into contact, energy transfer occurs until they reach a state of thermal equilibrium, where their temperatures equalize.
| Structural Concept / Function Type | Exact Mathematical Domain / Sign Properties | Active Operation Formulas / Identities | Systemic Role & Evaluation Limits |
|---|---|---|---|
| Heat Exchange ($Q$) | $Q > 0$ for heat absorption; $Q < 0$ for heat rejection. | $Q = m s \Delta T$ | Quantifies thermal energy transfer; limited to non-phase-change regimes. |
| Temperature ($T$) | $T \ge 0\text{ K}$ (Absolute scale limit) | $T = t_C + 273.15$ | Determines thermal state and direction of spontaneous heat flow. |
Temperature measurement relies on thermometric properties —physical characteristics of materials that vary linearly or predictably with temperature, such as liquid volume, electrical resistance, or gas pressure.
Standard scales are defined using two fixed reference points: the ice point ($0^\circ\text{C}$ or $32^\circ\text{F}$) and the steam point ($100^\circ\text{C}$ or $212^\circ\text{F}$).
The conversion formula between the Celsius ($t_C$) and Fahrenheit ($t_F$) scales is derived from their linear relationship:
$$\frac{t_F - 32}{180} = \frac{t_C}{100}$$
Gas thermometers provide highly consistent readings because low-density gases exhibit uniform expansion behavior regardless of their chemical composition. This behavior is governed by the ideal-gas equation :
$$PV = \mu R T$$
Where $\mu$ is the number of moles and $R = 8.31\text{ J mol}^{-1}\text{ K}^{-1}$ is the universal gas constant .
Historical Chronicle: Boyle and Charles
Most substances expand when heated and contract when cooled due to changes in intermolecular spacing. This phenomenon is categorized into linear, area, and volume expansion.
$$\frac{\Delta l}{l} = \alpha_l \Delta T$$
where $\alpha_l$ is the coefficient of linear expansion .$$\frac{\Delta A}{A} = \alpha_a \Delta T = 2\alpha_l \Delta T$$
$$\frac{\Delta V}{V} = \alpha_v \Delta T = 3\alpha_l \Delta T$$
where $\alpha_v$ is the coefficient of volume expansion .Water exhibits a unique, anomalous behavior: it contracts when heated from $0^\circ\text{C}$ to $4^\circ\text{C}$. Consequently, water reaches its maximum density at exactly $4^\circ\text{C}$ ( 1000 kg/m³ ).
This property is ecologically vital, as it prevents lakes and ponds from freezing solid from the bottom up, preserving aquatic life during winters.
If a structural member (like a steel rail or beam) is prevented from expanding or contracting by rigid supports, internal forces develop, creating thermal stress :
$$\text{Thermal Stress} = \frac{F}{A} = Y \alpha_l \Delta T$$
Where $Y$ is the Young's modulus of the material.
Mathematical Problem Statement: An iron ring of diameter 5.231 m at 27 °C is to be fitted onto a wooden wheel rim of diameter 5.243 m. To what temperature must the ring be heated? (Take $\alpha_l = 1.20 \times 10^{-5}\text{ K}^{-1}$ for iron)
Step-by-Step Algorithmic Solution Trace:
- Identify initial parameters: $L_{T1} = 5.231\text{ m}$, $L_{T2} = 5.243\text{ m}$, $T_1 = 27^\circ\text{C}$.
- Apply the linear expansion formula: $L_{T2} = L_{T1} [1 + \alpha_l (T_2 - T_1)]$.
- Isolate the temperature difference: $T_2 - T_1 = \frac{L_{T2} - L_{T1}}{L_{T1} \alpha_l} = \frac{5.243 - 5.231}{5.231 \times 1.20 \times 10^{-5}} \approx 191^\circ\text{C}$.
- Calculate final temperature: $T_2 = 191 + 27 = 218^\circ\text{C}$.
Final Evaluation Output: $T_2 = 218^\circ\text{C}$
The thermal response of a substance to heat absorption depends on its mass, temperature change, and intrinsic material properties.
$$S = \frac{\Delta Q}{\Delta T}$$
$$s = \frac{1}{m}\frac{\Delta Q}{\Delta T}$$
SI Unit: J kg⁻¹ K⁻¹ . Water has an exceptionally high specific heat capacity ( 4186 J kg⁻¹ K⁻¹ ), making it an excellent coolant.$$C = \frac{1}{\mu}\frac{\Delta Q}{\Delta T}$$
For gases, this is defined under constant pressure ($C_p$) or constant volume ($C_v$).Calorimetry is the quantitative measurement of heat exchange. In an isolated system, the law of conservation of energy dictates:
$$\text{Heat Lost} = \text{Heat Gained}$$
Measurements are performed in a calorimeter, a vessel insulated from the environment to minimize heat leakage.
Matter transitions between solid, liquid, and gaseous phases when heat is exchanged. During a phase transition, the temperature of the system remains constant despite continuous heat input or output.
The heat energy absorbed or released per unit mass during a phase change is called latent heat ($L$):
$$Q = m L$$
Heat energy transfers spontaneously from regions of higher temperature to regions of lower temperature via three distinct physical mechanisms: conduction, convection, and radiation.
Conduction is the transfer of heat through molecular collisions without bulk motion of the material. In a steady state, the rate of heat flow $H$ through a bar of length $L$ and cross-sectional area $A$ is:
$$H = KA \frac{T_C - T_D}{L}$$
Where $K$ is the thermal conductivity of the material (SI Unit: W m⁻¹ K⁻¹ ).
Convection is heat transfer by the actual movement of fluid matter. It can be natural (driven by buoyancy forces due to density differences, e.g., sea breezes, trade winds) or forced (driven by pumps or fans, e.g., blood circulation by the heart).
Radiation is heat transfer via electromagnetic waves, requiring no material medium.
$$\lambda_m T = \text{constant} \approx 2.9 \times 10^{-3}\text{ m K}$$
$$H = A \sigma T^4$$
For a real body with emissivity $e$ ($0 < e < 1$):$$H = e \sigma A T^4$$
Where $\sigma = 5.67 \times 10^{-8}\text{ W m}^{-2}\text{ K}^{-4}$ is the Stefan-Boltzmann constant.$$H_{\text{net}} = e \sigma A (T^4 - T_s^4)$$
According to Newton's law of cooling , the rate of loss of heat of a hot body is directly proportional to the difference in temperature between the body and its surroundings, provided the temperature difference is small.
Let a body of mass $m$ and specific heat capacity $s$ be at temperature $T_2$, and its surroundings be at temperature $T_1$. The rate of heat loss is:
$$-\frac{dQ}{dt} = k(T_2 - T_1)$$
Since $dQ = m s dT_2$, we can write:
$$-m s \frac{dT_2}{dt} = k(T_2 - T_1)$$
$$\frac{dT_2}{T_2 - T_1} = -\frac{k}{ms} dt = -K dt$$
Integrating both sides yields:
$$\ln(T_2 - T_1) = -Kt + C$$
$$T_2 = T_1 + C' e^{-Kt}$$
This exponential decay indicates that the cooling rate is initially rapid and slows down asymptotically as the body's temperature approaches that of its surroundings.