Thermal Expansion Concept Page - 1

Diagram

Heating curve of water

Heating curve of water is shown in the attached graph. It is assumed that heat is supplied at a constant rate.
First phase: Water at temperature <100o C$<{100}^{o}C$ to water at temperature  100o C${100}^{o}C$
Second phase: Water at 100o C${100}^{o}C$ to steam at temperature 100o C${100}^{o}C$.
Third phase: Steam at 100o C${100}^{o}C$ to steam at temperature >100o C$>{100}^{o}C$

Formula

Linear expansion in solid

Linear expansion in solids is given by the formula:
L2=L1(1+α(T2−T1))$L_2=L_1(1+\alpha (T_2-T_1))$
where
α:$\alpha :$ coefficient of the linear expansion
L1:$L_1:$ Length of solid at temperature T1$T_1$
L2:$L_2:$ Length of solid at temperature T2$T_2$ 

Example

Coefficient of linear expansion of different solids

Some values of coefficients of thermal expansion are:

Material	Coefficient of linear expansion (10−6 K−1${10}^{-6}{K}^{-1}$)
Mercury	60
Aluminium	23
Brass	19
Stainless Steel	17.3
Copper	17
Gold	14
Steel/Iron	11.1
Glass	8.5
Silicon	3
Diamond	1
Quartz, fused	0.59

Example

Percentage change in length due to thermal expansion

A thin rod of negligible mass and cross-sectional area 4×10−6m2$4\times {10}^{-6}{m}^2$, suspended vertically from one end, has a length of  0.5 m$0.5m$ at 100oC${100}^{o}C$. The rod is cooled to 0oC$0^{o}C$.
Young's modulus =1011 Nm−2$={10}^{11}N{m}^{-2}$, coefficient of linear expansion =10−5 K−1$={10}^{-5}{K}^{-1}$ and g=10 ms−2$g=10m{s}^{-2}$, ΔT=−1000C$\mathrm{\Delta }T=-{100}^{0}C$
The decrease in the length of the rod on cooling is:ΔL=LαΔT$\mathrm{\Delta }L=L\alpha \mathrm{\Delta }T$ =(0.5)(10−5)(−100)$=(0.5)({10}^{-5})(-100)$
ΔL=−5×10−4 m$\mathrm{\Delta }L=-5\times {10}^{-4}m$

Definition

Superficial expansion in solids

Expansion in area of a Laminar surface due to heating is known as superficial expansion.A metal sheet having size of 0.6 X 0.5 m2$m^2$ is heated from 293∘${293}^{\circ }$K to 520∘${520}^{\circ }$C. The final area of the hot sheet is
[αmetal=2×10−5/∘$[{\alpha }_{metal}=2\times {10}^{-5}{/}^{\circ }$C]In this case, β=4$\beta =4$×$\times$10−5${10}^{-5}$
A1$A_1$ =$=$ 0.6 X 0.5 =$=$  0.30 m2$m^2$
293∘${293}^{\circ }$K =$=$ 20∘${20}^{\circ }$ C
So, t2−t1=dt=520−20=500$t_2-t_1=dt=520-20=500$
Putting all the values in
A2=A1(1+βdt)$A_2=A_1(1+\beta dt)$
A2=0.30(1+4$A_2=0.30(1+4$X10−5${10}^{-5}$(500))
We get A2=0.306$A_2=0.306$ m2$m^2$

Formula

Superficial expansion coefficient

Coefficient of superficial expansion is defined as fractional change in the area per degree rise in temperature.
A2=A1(1+β(T2−T1))$A_2=A_1(1+\beta (T_2-T_1))$
where:
β:$\beta :$ Coefficient of superficial expansion
A1:$A_1:$ Area at temperature T1$T_1$
A2:$A_2:$ Area at temperature T2$T_2$

Example

Percentage change in area due to thermal expansion

When a thin rod of length l$l$ is heated from t∘1C$t_1^{\circ }C$ to t∘2C$t_2^{\circ }C$ length increases by 1%$1\mathrm{\%}$. If plate of length 2l$2l$ and breadth l$l$ made of same material is heated form t∘1C$t_1^{\circ }C$ to t∘2C$t_2^{\circ }C$, what will be the percentage increase in area? We know, l′=l(1+αdt)$l^{\prime }=l(1+\alpha dt)$ where dt = t2−t1$t_2-t_1$ degree centigrade
Or, (l′−l)/l=αdt$(l^{\prime }-l)/l=\alpha dt$
Percentage change in length = [(l′−l)/l]$[(l^{\prime }-l)/l]$×100=αdt×100=1$\times 100=\alpha dt\times 100=1$
Or, α$\alpha$ = 1 / (100 dt)
Or β$\beta$ = 2 / (100 dt)
Percentage change of area will be,
[(A′−A)/A]×100=βdt×100=2$[(A^{\prime }-A)/A]\times 100=\beta dt\times 100=2$ (putting value of β$\beta$)

Definition

Isotropic solids on heating

An isotropic solid is a solid material in which physical properties do not depend on its orientation.An isotropic solid will expand equally in all directions when heated and sound passes at the same speed in each direction.For isotropic solids the heat flux at a point is directly proportional to the temperature gradient at that point and the heat flux vector is normal to the isothermal surface passing through that point.

Definition

Relation between coefficients of linear and aerial expansion for isotropic solids

Length increases as
L′→L(1+αΔT)$L^{\prime }\to L(1+\alpha \mathrm{\Delta }T)$
But this means that for isotropic (same in every direction) expansion a surface (length x length) increases as
A′→A(1+αΔT)(1+αΔT)→A(1+2αΔT)$A^{\prime }\to A(1+\alpha \mathrm{\Delta }T)(1+\alpha \mathrm{\Delta }T)\to A(1+2\alpha \mathrm{\Delta }T)$
where we have neglected the (usually very small) square term ( T) .
Comparing with the (definition of ) expression
A(1+βΔT)$A(1+\beta \mathrm{\Delta }T)$ , we see the relation
β=2α$\beta =2\alpha$ . 
where  α$\alpha$ is the coefficient of linear expansion and β$\beta$ is the coefficient of aerial expansion

Formula

Volume Expansion of solid

Coefficient of cubical expansion is defined as fractional change in the volume per degree rise in temperature.
V2=V1(1+γ(T2−T1))$V_2=V_1(1+\gamma (T_2-T_1))$
where
γ:$\gamma :$ Coefficient of volume expansion
V2:$V_2:$ Volume at temperature T2$T_2$
V1:$V_1:$ Volume at temperature T1

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