Thermal Expansion & Heat Coefficients

Linear Thermal Expansion

The coefficient of linear thermal expansion \( \alpha \) quantifies how much a material expands or contracts when its temperature changes.

\[ \Delta L = \alpha \, L_0 \, \Delta T \]

\( \Delta L \) — change in length (m)
\( L_0 \) — original length (m)
\( \Delta T \) — temperature change (°C or K)
\( \alpha \) — linear expansion coefficient (1/°C)

Typical Values for Solids

Material	\( \alpha \) (×10^-6 /°C)
Aluminum	23
Copper	17
Iron	12
Steel (carbon)	11
Glass (Pyrex)	3.3
Concrete	12

The coefficient of volumetric thermal expansion \( \beta \) indicates how much the volume of a material changes with temperature.

\[ \Delta V = \beta \, V_0 \, \Delta T \]

For isotropic solids, \( \beta \approx 3\alpha \). For liquids, \( \beta \) must be obtained from experimental tables.

Volumetric coefficient of a solid container (isotropic solid):

\[ \beta_{\text{container}} = 3\alpha \]

\( \beta_{\text{container}} \) — volumetric expansion coefficient of the solid container (1/°C)
\( \alpha \) — linear expansion coefficient of the solid (1/°C)
Valid for isotropic solids with small temperature changes

¿Qué es un sólido isotrópico?

Solids

Material	\( \beta \) (×10^-6 /°C)
Aluminum	69
Copper	51
Iron	36
Steel (carbon)	33
Glass (Pyrex)	9.9
Concrete	36

Liquids

Liquid	\( \beta \) (×10^-5 /°C)
Ethanol	75
Water (20 °C)	21
Glycerin	49
Gasoline	95
Mercury	18

Example: \( \beta_{\text{ethanol}} = 75 \times 10^{-5} \, ^\circ\text{C}^{-1} \)

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Volumetric Thermal Expansion

The coefficient of volumetric thermal expansion \( \beta \) indicates how much the volume of a material changes with temperature.

\[ \Delta V = \beta \, V_0 \, \Delta T \]

For isotropic solids, \( \beta \approx 3\alpha \). For liquids, \( \beta \) must be obtained from experimental tables.

Solids

Material	Linear Expansion Coefficient \( \alpha \) (\( \times 10^{-6} / ^\circ\text{C} \))	Volumetric Expansion Coefficient \( \beta \) (\( \times 10^{-6} / ^\circ\text{C} \))
Aluminum	24	72
Brass and Bronze	19	60
Copper	17	51
Glass (Common)	9	27
Glass (Pyrex)	3.2	9.6
Invar	0.9	2.7
Fused Quartz	0.4	1.2
Lead	29	87
Steel	11	33
Concrete	12	36
Ice	52	156

Liquids

Liquid	Volumetric Expansion Coefficient \( \beta \) (\( \times 10^{-5} / ^\circ\text{C} \))
Ethanol	75
Carbon Disulfide	115
Glycerin	49
Mercury	18
Water	21
Gasoline	96

Specific Heat Capacity

In introductory calorimetry, the specific heat of water is treated as a standard constant. When working in calories:

\[ c_{\text{water}} = 1 \; \frac{\text{cal}}{\text{g·°C}} \qquad \text{or} \qquad c_{\text{water}} = 4186 \; \frac{\text{J}}{\text{kg·K}} \]

Unit conversion

The joule in terms of base SI units:

\[ J = N \cdot m = \frac{kg \cdot m^2}{s^2} \]

The two units for specific heat capacity are related by the definitions \(1\,\text{cal} = 4{,}186\,\text{J}\) and \(1\,\text{g} = 0{,}001\,\text{kg}\), which together give a conversion factor of 4186:

\[ \frac{J}{kg \cdot K} \;\xrightarrow{\div\,4186}\; \frac{\text{cal}}{g \cdot °C} \qquad \frac{\text{cal}}{g \cdot °C} \;\xrightarrow{\times\,4186}\; \frac{J}{kg \cdot K} \]

Example with water:

\[ 4186 \;\frac{J}{kg \cdot K} \;\div\; 4186 \;=\; 1 \;\frac{\text{cal}}{g \cdot °C} \]

J/kg·K vs J/kg·°C: these are identical. A temperature difference of 1 K equals 1 °C (\(\Delta T_K = \Delta T_{°C}\)), so the two units are interchangeable:

\[ 1 \;\frac{J}{kg \cdot K} = 1 \;\frac{J}{kg \cdot °C} \]

Importante: si en un calculo se mezclan unidades — por ejemplo \(Q\) en calorías pero \(P\) en watts \(\bigl(\tfrac{J}{s}\bigr)\) — hay que convertir antes de dividir, de lo contrario el resultado queda desviado por un factor de 4186.

Common Liquids

Liquid	\( c \) (cal / g·°C)	\( c \) (J / kg·K)
Water	1.00	4186
Ice	0.45	1884
Ethanol	0.58	2428
Glycerin	0.60	2512
Mercury	0.03	126
Vegetable Oil	0.50	2093
Aluminum	0.215	900
Copper	0.092	385
Iron	0.107	448
Steel	0.12	502
Lead	0.031	128

Latent Heat

Latent heat is the energy absorbed or released during a phase transition at constant temperature. There are two types:

Heat of fusion \( L_f \): solid ↔ liquid
Heat of vaporization \( L_v \): liquid ↔ gas

\[ Q = m \, L \]

Where:

\(Q\) — heat absorbed or released during the phase transition (in joules, \(J\))
\(m\) — mass of the substance undergoing the phase change (in kilograms, \(kg\))
\(L\) — latent heat of the substance, either \(L_f\) (fusion) or \(L_v\) (vaporization) (in \(J/kg\))

Water / Ice

Transition	\( L \) (cal / g)	\( L \) (kJ / kg)
Fusion (ice → water, 0 °C)	80	334
Vaporization (water → steam, 100 °C)	540	2260

Signo de \(L\)

Los valores de la tabla indican magnitudes. En la fórmula \(Q = mL\), el signo de \(L\) depende del proceso:

\(L > 0\) — el proceso absorbe calor del entorno: fusión, vaporización, sublimación.
\(L < 0\) — el proceso cede calor al entorno: solidificación, condensación, deposición.

Physical Constants

Universal Gas Constant \( R \)

\[ R \approx 8{,}314 \, \frac{J}{mol \cdot K} \]

Used in the ideal gas equation \( pV = nRT \) and in thermal transformation formulas (isobaric, isothermal).

Adiabatic Coefficient \( \gamma \)

Gas type	\( \gamma \)	Examples
Monatomic	\( \tfrac{5}{3} \approx 1{,}67 \)	He, Ar, Ne
Diatomic	\( \tfrac{7}{5} = 1{,}4 \)	N₂, O₂, H₂, air