Thermal Expansivity
Physics SS1 Second Term
WEEK 2
Thermal Expansivity
Performance Objectives
Students should be able to:
- Explain Linear expansivity
- Area Expansivity
- Volume or Cubic Expansivity
Content
When a solid is heated is to a particular temperature, the molecule it contains absorb some kinetic energy since kinetic energy is the measure of velocity (i.e. the higher the kinetic energy, the higher the velocity of the molecules). These molecules move faster and collide with one another and with the wall of the system to the event that the strong molecular force of attraction is broken after several collisions and by so doing the system expands. Linear expansivity of solid (metal) is defined as the increase in length per unit rise in temperature. It is denoted by α and measured in K-1 or 0C-1 (SI unit).
Original length at to1C = Lo
Final length when heated at to1 C = Lo
Final length when heated at to2C = Lt
Linear expansivity/coefficient of expansion = α
Change in temperature from t2-t1 =∆ᶿ.
α= L2-L1 / L1∆ᶿ
L2-L1 = L1α∆ᶿ
L2 = L1 (1 +α∆ᶿ)
Example: If a metal rod of length 50m expands when heated at a temperature change of 10k, find the charge in length (linear expansivity of metal), [α= 1.5 x 10-8 k-1]
Solution:
Using α= L2-L1 / L1∆t given that L1 = 50m, t = 10k
∆L = L2-L1 = L1∆t α= 1.5 x 10-8 k-1
∆L = L2-L1 = 1.5 x 10-8 x 10 x 50
∆L = L2-L1 = 750 x 10-8
∆L = L2-L1 = 7.5 x 10-6m
Change in length is 7.5 x 10-6m
Area and Volume expansivity
When a solid is heated, it expands in all directions-in length, in breath and in height. Hence there is an increase in the area as well as in the volume of the solid. The increase in area when a body is heated is known as area or superficial expansion.
The area or superficial expansivity, β, of a solid is the increase in area per unit area degree Kelvin increase in temperature or the fractional increase in area per Kelvin rise in temperature.
Similarly, an increase in volume when a body is heated is known as cubic or volume expansion and we define expansivity as follows:
The volume or cubic expansivity, y, is the increase in the volume of a substance per unit volume per Kelvin rise in temperature or the fractional increase in volume per Kelvin rise in temperature.
Area expansivity β
= change in area/original area x temperature rise
β= A2-A1 / A1 x ᶿ
where A2 = area at temperature ᶿ2
A1 = area at temperature ᶿ1
ᶿ= ᶿ2-ᶿ1
A2 = A1(1 +βᶿ)
Increase in area = A2-A1=A1 βᶿ
Volume Thermal Expansion
When the temperature of a volume changed ∆T, the change of its volume ∆V is very nearly proportional to its initial volume multiplied by ∆T. The Volume Expansion equation is:
∆V = βV0∆T
Where:
β: the Coefficient of volume expansion
V0: Initial volume of the object
∆V: Volume change of the object
∆T: Temperature change of the object
We have seen that change in length (or volume) with changing temperature can be used to create a thermometer. The old-fashioned mercury in glass thermometer relies on this phenomenon.
But exactly how much does an object change its length (or volume) as the temperature changes? Experimentally we find that for a solid or liquid, the length change is given by,
∆l = αl0∆T
Where l is the ‘original’ length, ∆T is the change in temperature and α is the coefficient of linear expansion. The coefficient of linear expansion is approximately independent of temperature and is very small, e.g. for steel α= 12 x 10-6 K-1.
Linear expansion is important only when one dimension of an object is much larger than the other two. For volume expansion it can be shown that,
∆V = βV0∆T
where V0 is the ‘’original’’ volume and β = 3αis the volume coefficient of expansion.
Anomalous Expansion of Water
Water is a unique material in many ways. Without it and its unusual properties life as we know it could not exist. One of these critical properties is the so-called anomalous expansion. When heated, most substances expand according to the simple description above, but when heated through the temperature range 0-4 0C water contracts (its density increases).
Furthermore, as you are probably aware when water is cooled and freezes (at 0 0C) the frozen water (ice) expands becoming less dense, causing problems with roads, buildings etc. This property is very unusual, most substances, on freezing contract rather than expand. This property of water explains why ice floats. If this were not the case, the ice created in the winter would sink to the bottom of lakes and oceans and would be insulated from melting in the summer. After a short time, our oceans and lakes would consist of a solid mass of ice with only a thin layer at the surface alternately freezing and melting as the seasons came and went-hardly conducive to the development of life.
Expansion of Liquid
Liquid Expansion
Real expansivity of liquid
Real expansivity of liquid is sometimes called cubic expansivity of liquid and it is defined as the increase in volume per unit degree rise in temperature. Since liquid does not have a particular length or area, then we talk of its volume about its container.
The cubic expansivity is sometimes called the real expansivity yr
When a liquid is heated in a vessel, expansion occurs both in the liquid and in the vessel. Then from the vessel, we have apparent expansivity of the liquid.
Apparent cubic expansivity
The apparent cubic expansivity of a liquid is defined as the mass of the liquid expelled per unit divided by mass left or remaining when the temperature increases by 1oC. It is measured in K-1.
Thus,
Apparent cubic expansivity = mass of liquid expelled/mass of liquid left x temperature rise.
Expansion of various liquids