# Thermal Expansion

Written by Jerry Ratzlaff on . Posted in Thermodynamics

## Thermal Expansion

The increase in length, area or volume due to the increase (in some cased decrease) in temperature.  The stored energy in the molecular bonds between atoms changes when the heat transfer occurs.  The length of the molecular bond increases as the stored energy increases.

Area thermal expansion - Area measures length times width and is two dimensional so it is squared.  Expands twice as much as lengths do.

Linear thermal expansion - Linear measures distance or length and is one dimensional.  Can only be measured in the solid state.  The expansion is proportional to temperature change.

Volumetric thermal expansion - Volume measures length times width times height so it is three dimensional so it is cubed.  Can be measured for all substances (liquid or solid) of condensed matter.  Expands three times as much as lengths do.

Some substances such as water can increase or decrease depending on the temperature.

## Area Thermal Expansion

Area thermal expansion (also known as aerial thermal expansion) happens when any change in temperature expands the area.

### Area Thermal Expansion Formulas

$$\large{ \Delta A = A_f \;-\; A_i }$$

$$\large{ \Delta A = 2 \alpha \Delta T A_i }$$

$$\large{ \Delta A = A_i \left( 1 + 2 \alpha \Delta T \right) }$$

$$\large{ \Delta A = \gamma A_i \Delta T }$$

$$\large{ \frac { \Delta A } { A_i } = \gamma \Delta T }$$

Where:

$$\large{ \alpha }$$   (Greek symbol alpha) = linear thermal expansion coefficient

$$\large{ A_f }$$ = final area of object

$$\large{ A_i }$$ = initial area of object

$$\large{ \Delta A }$$ = area differential

$$\large{ \Delta T }$$ = temperature differential

$$\large{ \gamma }$$   (Greek symbol gamma) = area thermal expansion coefficient

## Area Thermal Expansion Coefficient

Area thermal expansion coefficient ( $$\gamma$$ ) (also known as coefficient of aerial thermal expansion) is the ratio of the change in size of a material to its change in temperature.

### Area Thermal Expansion Coefficient FORMULA

$$\large{ \gamma = \frac { 1 }{ A } \frac {\Delta A } {\Delta T} }$$

Where:

$$\large{ \gamma }$$   (Greek symbol gamma) = area thermal expansion coefficient

$$\large{ A }$$ = area of the object

$$\large{ \Delta A }$$ = area differential

$$\large{ \Delta T }$$ = temperature differential

## Linear Thermal Expansion

Linear thermal expansion (also known as line thermal expansion) is a porportional change in the origional length and change in temperature due to the heating or cooling of an object.

### Linear Thermal Expansion Formulas

$$\large{ \Delta l = l_f \;-\; l_i }$$

$$\large{ \Delta l = \alpha \Delta T l }$$

$$\large{ l_f = l_i \left( 1 + \alpha \Delta T \right) }$$

$$\large{ \frac { \Delta l } { l_i } = \alpha \Delta T }$$

Where:

$$\large{ \alpha }$$   (Greek symbol alpha) = linear thermal expansion coefficient

$$\large{ l_f }$$ = final length

$$\large{ l_i }$$ = initial length

$$\large{ \Delta l }$$ = length differential

$$\large{ T }$$ = temperature

$$\large{ \Delta T }$$ = temperature differential

## Linear Thermal Expansion Coefficient

Linear thermal expansion coefficient ( $$\alpha$$ ) (also known as coefficient of linear thermal expansion) is the ratio of the change in size of a material to its change in temperature.

### Linear Thermal Expansion Coefficient FORMULA

$$\large{ \alpha_l = \frac { 1 }{ l } \frac {\Delta l } {\Delta T} }$$

Where:

$$\large{ \alpha }$$   (Greek symbol alpha) = linear thermal expansion coefficient

$$\large{ l }$$ = length of the object

$$\large{ \Delta l }$$ = Length differential

$$\large{ \Delta T }$$ = temperature differential

## Linear Thermal Restrained Expansion

### Linear Thermal Restrained Expansion Formulas

$$\large{ \Delta p = \lambda A_i \alpha \Delta T }$$

$$\large{ \sigma_c = \;-\; \frac{p}{A_i} }$$

$$\large{ \sigma_c = \;-\; \lambda A_i \alpha \Delta T }$$

Where:

$$\large{ \alpha }$$   (Greek symbol alpha) = linear thermal expansion coefficient

$$\large{ A_i }$$ = initial area of object

$$\large{ p }$$ = pressure

$$\large{ \Delta p }$$ = pressure differential

$$\large{ \Delta T }$$ = temperature differential

$$\large{ \lambda }$$  (Greek symbol lambda) = modulus of elasticity

$$\large{ \sigma_c }$$  (Greek symbol sigma) = compressive stress

## Thermal Expansion Coefficient

Thermal expansion coefficient also called coefficient of thermal expansion, and linear thermal expansion coefficient, is the percentage change in the length of the material per degree of temperature change, heated solid or liquid.

• $$\alpha$$ (Greek symbol alpha) = thermal expansion coefficient

## Thermal Expansion of Gases

### Thermal Expansion of Gases Formula

$$\large{ pV = n R T }$$ = ideal gas law

Where:

$$\large{ n }$$ = number of moles of gas

$$\large{ p }$$ = pressure

$$\large{ R }$$ = specific gas constant (gas constant)

$$\large{ T }$$ = temperature

$$\large{ V }$$ = volume

## Thermal Expansion of Liquids

### Thermal Expansion of Liquids Formula

$$\large{ \Delta V = \beta V_i \Delta T }$$ = volumetric or cubical expansion

Where:

$$\large{ \beta }$$   (Greek symbol beta) = volumetric thermal expansion coefficient

$$\large{ \Delta T }$$ = temperature differential

$$\large{ V_i }$$ = initial volume of object

$$\large{ \Delta V }$$ = volume differential

## Thermal Expansion of Solids

### Thermal Expansion of Solids Formulas

$$\large{ \Delta l = \alpha l_i \Delta T }$$ = linear expansion

$$\large{ \Delta A = 2 \alpha A_i \Delta T }$$ = aerial or superficial expansion

$$\large{ \Delta V = 3 \alpha V_i \Delta T }$$ = volumetric or cubical expansion

Where:

$$\large{ \alpha }$$   (Greek symbol alpha) = linear thermal expansion coefficient

$$\large{ A_i }$$ = initial area of object

$$\large{ \Delta A }$$ = area differential

$$\large{ l_i }$$ = initial length

$$\large{ \Delta l }$$ = length differential

$$\large{ \Delta T }$$ = temperature differential

$$\large{ V_i }$$ = initial volume of object

$$\large{ \Delta V }$$ = volume differential

## Volumetric Thermal Expansion

Volumetric thermal expansion (also known as volume thermal expansion) takes place in gasses and liquids when a change in temperature, volume or type of substance occures.

### Volumetric Thermal Expansion Formulas

$$\large{ \Delta V = V_f \;-\; V_i }$$

$$\large{ \Delta V = 3 \alpha \Delta T V }$$

$$\large{ \Delta V = V \left( 1 + 3 \alpha \Delta T \right) }$$

$$\large{ \Delta V = \beta V_i \Delta T }$$

$$\large{ \frac { \Delta V} { V_i } = \beta \Delta T }$$

Where:

$$\large{ \alpha }$$   (Greek symbol alpha) = linear thermal expansion coefficient

$$\large{ \beta }$$   (Greek symbol beta) = volumetric thermal expansion coefficient

$$\large{ \Delta T }$$ = temperature differential

$$\large{ V }$$ = volume

$$\large{ V_f }$$ = final volume of object

$$\large{ V_i }$$ = initial volume of object

$$\large{ \Delta V }$$ = volume differential

## volumetric Thermal Expansion Coefficient

Volumetric thermal expansion coefficient ( $$\beta$$ ) (also known as coefficient of volumetric thermal expansion) is the ratio of the change in size of a material to its change in temperature.

### Volumetric Thermal Expansion Coefficient FORMULA

$$\large{ \beta = \frac { 1 }{ V } \frac {\Delta A } {\Delta T} }$$

Where:

$$\large{ \beta }$$   (Greek symbol beta) = volumetric thermal expansion coefficient

$$\large{ \Delta A }$$ = area differential

$$\large{ \Delta T }$$ = temperature differential

$$\large{ V }$$ = volume of the object