# Thermal Expansion

In general, all bodies expand when they are heated and contract when cooled. The expansion or contraction, as the case may be, is small in solids, larger in liquids, and the largest in gases. The increase in length, width or height of a solid is called linear expansion. The increase in area is called surface expansion. The increase in volume is called volume expansion or cubical expansion. A liquid or a gas has only volume expansion.

Suppose the length of a rod increases from an initial value of l1 at temperature θ1 to a final value of l2 at higher temperature θ2. The relation between these two lengths is:

` l2 = l1 [1 + α (θ2 - θ1)] `

Here α is a constant of proportionality, known as the coefficient of linear expansion for the solid material of the rod. The dimension of α is [K-1] , and its unit is per kelvin (/K).

In a similar way, we can relate the surface areas of a solid plate at two different temperatures as:

``` S2 = S1 [1 + β (θ2 - θ1)] ```

where β is the coefficient of surface expansion for the solid. And for change in volume with temperature:

``` V2 = V1 [1 + γ(θ2 - θ1)] ```

The quantity, γ, is the coefficient of volume expansion. All these three coefficients have the same dimension and the same unit. And for an isotropic solid, a simple relation exists among them:

``` α = β/2 = γ/3 ```

Thus, referring to one table for the values of α of various solids, we can determine the corresponding values of β and γ.

Some notable effects of thermal expansion of solids are expansion of a measuring scale, a faulty pendulum clock that runs slow in summer, thermal stress experienced by a rod fixed between two rigid walls, deformation of a bimetallic strip that is cleverly used in a thermostat to switch an air-conditioner on and off.

While discussing the expansion of liquids, it must be kept in mind that both the liquid and the solid container expand in volume as temperature goes up. So it is important to distinguish between apparent expansion and real expansion of a liquid. Correspondingly, there would be two coefficients for a liquid: coefficient of apparent expansion and coefficient of real expansion. The relation between these two coefficients is:

``` γr = γa + γg ```

The symbols are nearly self-explanatory; γg is the coefficient of volume expansion of glass or any other solid material from which the container is made. The coefficients of real expansion of liquids are significantly greater than the coefficients of volume expansion of solids.

Some devices to measure the coefficients of expansion of a liquid are dilatometer or volume thermometer, weight thermometer, Dulong and Petit's apparatus etc.

One notable effect of thermal expansion of liquids is the change in buoyant force on an immersed body with change in temperature. Another is the correction factor to be introduced to the reading of a barometer if the ambient temperature is different from the temperature of calibration. Anomalous expansion of water refers to the strange phenomenon of water contracting in volume as its temperature rises from 0°C to 4°C, implying a negative expansion coefficient in that range!

Expansion of gases is dictated by three famous laws: Boyle's law, Charles' law and pressure law. The last two laws lead us to the concept of absolute zero of temperature, which is - 273.15°C or 0 K. This is the temperature at which, theoretically, a gas would occupy zero volume and exert zero pressure. In reality, all gases change into liquid phase much above absolute zero. Combining Boyle's law and Charles' law, we can arrive at the equation of state of an ideal gas which is:

` PV = nRT `

Here P is pressure and V is volume of n moles of gas at Kelvin temperature T. The quantity R is called universal gas constant, because its value is the same 8.31 J/mol-K for all gases. The ratio of universal gas constant and Avogadro's number is called Boltzmann's constant. The equation of state of a gas mixture can be derived by making use of Dalton's law of partial pressure.

#### Basic level videos

Thermal expansion of solids 1:37:16 Basic
400 4
Effects of thermal expansion of solids 1:02:06 Basic
400 3
Problems on thermal expansion of solids and its effects I 1:05:32 Basic
400 3
Problems On Thermal Expansion Of Solids And Its Effects II 1:36:03 Basic
500 5
Thermal expansion of liquids 1:08:35 Basic
400 4
Measurement Of Coefficients Of Expansion Of A Liquid 51:51 Basic
300 3
Effects Of Thermal Expansion Of Liquids 47:34 Basic
300 3
Problems On Thermal Expansion Of Liquids And Its Effects I 1:18:07 Basic
300 3
Expansion Of Gases 1:09:53 Basic
300 3
Expansion Of Gases II 41:57 Basic
300 3
Problems On Expansion Of Gases 1:41:12 Basic
300 3

#### Advanced level Videos Note: (CE) Stands for Problems from Competitive Examination Papers

Advanced-level Problems On Thermal Expansion I 1:09:39
400
4
Advanced-level Problems On Thermal Expansion II 1:21:59
300
3
Advanced-level Problems On Thermal Expansion III 1:10:26
300
3
Advanced-level Problems On Thermal Expansion IV 1:05:49
300
3