Thermal Expansion

 Thermometry

$\text{CONCEPTS: }$$\to$With the increase in temperature of a body, there is an increase in its dimension also. This property is known as thermal expansion. When temperature of body increases intermolecular separation also increases which result in thermal expansion of a body.$\\$$\to$We can see thermal expansion in all states (liquid,gas and solid). Solids have all kinds of expansion but liquid and gases only have volume expansion.

Types of Expansion In Solids

$\text{Linear Expansion: }$Increase in length of the body during heating.$\\$$L_2$=$L_1$(1+αLΔT), where $L_1$ and $L_2$ are the length before expansion and after expansion.$\\$α is coefficient of linear expansion, ΔT is change in temperature.$\\$Also, it can be represented by ΔL/L = αLΔT (ΔL = $L_2$-$L_1$)$\\$

$\textbf{Superficial Expansion: } \\$Increase in Area of the body during heating.$\\$$A_2$=$A_1$(1+βAΔT), where $A_1$ and $A_2$ are the area before expansion and after expansion.$\\$β is coefficient of Superficial expansion, ΔT is change in temperature.$\\$Also, it can be represented by ΔA/A = βAΔT (ΔA = $A_2$-$A_1$)$\\$

$\textbf{Cubical Expansion: } \\$Increase in Area of the body during heating.$\\$$V_2$=$V_1$(1+γ AΔT), where $V_1$ and $V_2$ are the area before expansion and after expansion.$\\$γ is coefficient of Cubical expansion, ΔT is change in temperature.$\\$Also, it can be represented by ΔV/V = γ VΔT (ΔV = $V_2$-$V_1$)$\\$

$\textbf{Notice:} \\$The relation between α, β and ϒ α:β:ϒ = 1:2:3. α, β and ϒ solely depends only upon nature of material of body

Expansion of Liquid

$\text{Expansion of Liquids:}$Liquids have only definite volume not definite shape. Its container determines its shape. When a liquid is heated in a vessel, both the liquid and a vessel expand.

The change in volume of liquid relative to vessel is called its apparent expansion.$\\$

The real change in volume of liquid only is called real expansion$\\$(without considering expansion of vessel). Thus there are two coefficients.$\\$Also, we can write: $ϒ_r$ = $ϒ_a$ + $ϒ_v$ Where $ϒ_r$ is the coefficient of real expansion, $ϒ_a$ is the coefficient of apparent expansion of liquid, and $ϒ_v$ is apparent expansion of vessel.

The apparent change in volume of liquid is given by ∆$V_a$ = V (ϒ - 3α)∆T.$\\$Where V is volume of liquid, ϒ is coefficient of volume expansion of liquid, α is coefficient of linear expansion of vessel.$\\$-If ϒ > 3 α, liquid level rises.$\\$-If ϒ < 3 α liquid level falls.$\\$-If ϒ = 3 α level of liquid remains stationary.

Temperature and Time Period

Δt =$\frac{1}{2}$αΔT (Change in time period)$\\$

($\frac{Δt}{t}$)*100=$\frac{1}{2}$ αΔT (Percentage change in time period)$\\$Temperature increases → Loss in time period$\\$Temperature decreases → Gain in time period

Anomalous Expansion

$\text{Anomalous expansion of water:}$

The anomalous expansion of water is an abnormal property of water$\\$whereby it expands instead of contracting when the temperature$\\$goes from $4^oC$ to $0^oC$. Water has maximum density (volume minimum) at 4.$\\$

The density becomes less and less as it freezes because molecules$\\$of water normally form open crystal structures when in solid form.

Bimetallic Strip

A strip consisting of two metals of different coefficients of expansion$\\$welded together so that it buckles on heating.$\\$

It is used as a thermostat and fire alarm.$\\$

When bimetallic strip is cooled it bends towards larger coefficient of linear expansion.

Thermal Stress

$\text{Thermal Stress:}$

When a metal rod is heated or cooled it expands or contracts. If it is prevented from$\\$expansion or contraction, then stress produced is thermal stress.$\\$

$\textbf{Strain}$ =$\frac{ΔL}{L}$= α ∆T$\\$

$\textbf{Thermal stress }$ = Y x strain = Y α ∆T

$\textbf{Force}$ = stress x A = Y α ∆T x A$\\$

∴ $\textbf{F}$ = YA α ∆T$\\$

$\textbf{Heating of Metal Plate with Hole}\\$

After expansion:$\\$R’ = R(1+α∆T) ….(i)$\\$r’ = r (1+α∆T) ….(ii)$\\$

From (i) and (ii)$\\$

R’.r=r’.R=constant

Memory Tips

Two identical spheres (solid and hollow) are heated:$\\$i)By giving equal heat then hollow expand more as it has less mass.$\\$ii)By same temperature then both will expand equally.$\\$

Two rods of length $L_1$ and $L_2$of linear expansivities $α_1$ & $α_2$ having$\\$same difference in temperature then $L_1$.$α_1$= $L_2$.$α_2$ .$\\$

Zeroth law of thermodynamics states that “If two bodies are thermal$\\$equilibrium separately with third body then these bodies are in thermal$\\$equilibrium with each other .”

Some More

$\text{Other tip’s:}$$\to$Gases have only volume expansion.$\\$$\to$When temperature of liquid increases its volume increases and density decreases.$\\$$\to$As density of substance decreases with increase in temperature so$\\$same volume of benzene weighs less in summer than in winter but same mass of$\\$benzene weighs equally in summer and winter (as mass is constant).$\\$$\to$When a metallic piece is weighed in a temperature increasing liquid then its apparent weight increases.$\\$$\to$Invar is used in pendulum clock (because of low coefficient of linear expansion.$\\$$\to$If C and S are the coefficient of apparent expansion of copper and silver respectively. The coefficient of linear expansion of copper is A than silver is given by $\frac{(C+3A-S)}{3}$$\\$$\to$The brass disc fits tightly in a hole in a steel plate. To lose the disc we should cool the system.$\\$$\to$The correction in height of mercury level in barometer due to expansion of mercury is given by : $H_{correct}$=$H_{scale}$[1+( α –r)∆T]$\\$

$\to$Density decreases with increase in temperature $ρ_t$= $ρ_0$(1-r∆T)$\\$

$\to$Apparent expansion of vessel depends upon nature of liquid, orginal$\\$volume of liquid, change in temperature and material of vessel.$\\$$\to$Real expansion doesn’t depend on expansion of vessel but depends upon$\\$temperature difference of liquid, initial volume of liquid and its nature.$\\$$\to$Water will boil when a bottle of water at zero degree Celsius opened on$\\$surface of moon because pressure of air on the surface of moon is almost$\\$zero.