Thermal expansion

Rosimar Gouveia

In general, bodies, whether solid, liquid or gaseous, increase in size as they increase in temperature.

Thermal Expansion of Solids

A rise in temperature causes the vibration and distance between the atoms that make up a solid body to increase. As a result, an increase in their dimensions occurs.

Depending on the most significant expansion in a given dimension (length, width and depth), the expansion of solids is classified as: linear, superficial and volumetric.

Linear Dilatation

Linear dilation takes into account the dilation suffered by a body in only one of its dimensions. This is the case, for example, with a thread, where its length is more relevant than its thickness,

To calculate linear dilation we use the following formula:

ΔL = L0.α.Δθ

Where,

ΔL : Length Change (m or cm)
L0 : Initial Length (m or cm)
α : Coefficient of linear expansion (ºC -1 )
Δθ : Temperature variation (ºC)

Surface Dilatation

Surface swelling takes into account the swelling of a given surface. This is the case, for example, with a thin metal plate.

To calculate surface expansion we use the following formula:

ΔA = A0.β.Δθ

Where,

ΔA : Area variation (m 2 or cm 2 )
A 0 : Initial area (m 2 or cm 2 )
β : Coefficient of surface expansion (ºC < sup> -1 )
Δθ : Temperature variation (° C)

It should be noted that the surface expansion coefficient (β) is equal to twice the value of the linear expansion coefficient (α), ie:

β = 2 . α

Volumetric Dilatation

Volumetric dilation results from an increase in body volume, such as a gold bar.

To calculate volumetric dilation we use the following formula:

ΔV = V0.γ.Δθ

Where,

ΔV : Volume change (m 3 or cm 3 )
V 0 : Initial volume (m 3 or cm 3 )
γ : Coefficient of volumetric expansion (ºC < sup> -1 )
Δθ : Temperature variation (° C)

Note that the volumetric expansion coefficient ( γ) is three times greater than the linear expansion coefficient (α), ie:

γ = 3 . α

Linear Dilatation Coefficients

The dilation a body undergoes depends on the material that composes it. Thus, in the calculation of the dilatation, the substance from which the material is made is taken into account through the linear expansion coefficient (α).

The table below indicates the different values ​​that can assume the linear expansion coefficient for some substances:

Substância Coeficiente de Dilatação Linear (ºC-1)
Porcelana 3.10-6
Vidro Comum 8.10-6
Platina 9.10-6
Aço 11.10-6
Concreto 12.10-6
Ferro 12.10-6
Ouro 15.10-6
Cobre 17.10-6
Prata 19.10-6
Alumínio 22.10-6
Zinco 26.10-6
Chumbo 27.10-6

Dilatação Térmica dos Líquidos

Os líquidos, salvo algumas exceções, aumentam de volume quando a sua temperatura aumenta, da mesma forma que os sólidos.

However, we must remember that liquids do not have their own form, acquiring the shape of the container that contains them.

Therefore, for liquids, it makes no sense to calculate either linear or superficial dilation, just volumetric.

Thus, we present below the table of the volumetric expansion coefficient of some substances.

Liquids Volumetric Dilatation Coefficients (ºC -1 )
Water 1,3.10 -4
Mercury 1.8.10 -4
Glycerin 4,9.10 -4
Alcohol 11,2.10 -4
Acetone 14,93.10 -4

Want to know more? Also read :

  • Heat and Temperature
  • Calorimetry
  • Thermometric Scales
  • Thermodynamics
  • Physics Formulas
  • Exercises

    1) A steel wire is 20 m long when its temperature is 40 ° C. What will be its length when its temperature is 100 ° C? Consider the coefficient of linear expansion of steel equal to 11.10 -6 ºC -1 .

    To find the final wire length, let's first calculate its variation for this temperature range. For this, simply replace in the formula:

    ΔL = L0.α.Δθ
    ΔL = 20.11.10-6.(100-40)
    ΔL = 20.11.10-6.(60)
    ΔL = 20.11.60.10-6
    ΔL = 13200.10-6
    ΔL = 0,0132

    To know the final size of the steel wire, we have to add the initial length with the variation found:

    L = L0+ΔL
    L = 20+0,0132
    L = 20,0132 m

    2) A square aluminum plate has equal sides at 3 m when its temperature is 80 ° C. What will be the variation in its area if the plate is subjected to a temperature of 100 ° C? Consider the linear expansion coefficient of aluminum 22.10 -6 ºC -1 .

    Como a chapa é quadrada, para encontrar a medida da área inicial devemos fazer:

    A0 = 3.3 = 9 m2

    The value of the linear expansion coefficient of aluminum was informed, however, to calculate the surface variation we need the value of β. So, first let's calculate this value:

    β = 2. 22.10 -6 ºC -1 = 44.10 -6 ºC

    We can now calculate the variation of the plate area by substituting the values ​​in the formula:

    ΔA = A0.β.Δθ
    ΔA = 9.44.10-6.(100-80)
    ΔA = 9.44.10-6.(20)
    ΔA = 7920.10-6
    ΔA = 0,00792 m2

    The Area Variation is 0.00792 m 2 .

    3) One 250 ml glass bottle contains 240 ml of alcohol at a temperature of 40 ° C. At what temperature will alcohol begin to overflow from the bottle? Consider the coefficient of linear expansion of the glass equal to 8.10 -6 ºC -1 and the volumetric coefficient of alcohol 11,2.10 -4 ºC < sup> -1 .

    First we need to calculate the volumetric coefficient of the glass, because only its linear coefficient was entered. Thus we have:

    γ Vidro = 3. 8 10 -6 = 24. 10 -6 ºC -1

    Both vial and alcohol dilate and alcohol will begin to overflow when its volume is larger than the vial volume.

    When both volumes are equal, the alcohol is about to overflow from the bottle. In this situation we have that the volume of alcohol is equal to the volume of the glass bottle, ie V glass = V alcohol .

    The final volume is found by making V = V 0 + ΔV. Substituting in the expression above, we have:

    V 0 glass + ΔV glass = V 0 + ΔV alcohol >

    Overriding problem values:

    250 + (250 . 24 . 10-6. Δθ) = 240 + (240 . 11,2 . 10-4. Δθ)
    250 + (0,006 . Δθ) = 240 + (0,2688 . Δθ)
    0,2688 . Δθ - 0,006 . Δθ = 250 - 240
    0,2628. Δθ = 10
    Δθ = 38 ºC

    To know the final temperature, we have to add the initial temperature with its variation:

    T=T0+ΔT
    T=40+38
    T=78 ºC

    Rosimar Gouveia