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In today’s class, we will be talking more about expansivity. Enjoy the class!
Expansivity II
Linear (coefficient of linear) expansivity
Different solids expand by different amounts when heated over the same temperature range. Copper for instants will expands more than steel when both are heated through the same rise in temperature. This is because they have a different coefficient of linear expansion or linear expansivity.
Linear expansivity is defined as the increase in length per unit length when the temperature of solid rises by one degree (1k)
Linear expansivity = l2 – l1 / l1(2 – 1) = e / l1
In symbols, it is equivalent to:
l1 = l2 – l1
l2 = l1 + l1
l2 = l1 (1 + )
Increase in length = l2 – l1
Where = linear expansivity
l2 = length of metal at temperature (1)
l1 = length of metal at temperature (2)
T = temperature rise which is given by (T2 – T1)
e =e2 – e1 = expansion or increase in length
The unit of is per °C or per K (K-1).
The statement that the linear expansivity of glass is 0.0000085 K-1 or 0.0000085/°C means that a unit length of glass expands by 0.0000085 units when it is heated through 1 K (1°rise in temperature.
Area (superficial) expansivity
When a solid is heated, it expands in all directions – in length and breadth. Hence there is an increase in the area of the solid. The increase in the area when a body is heated is known as an area of superficial expansion.
Therefore superficial expansion is defined as the increase in area per unit area per degree rise in temperature or is the fractional increase in area per Kelvin rise in temperature.
Note that for a given solid, the area expansivity is twice the linear expansivity.
Area Expansivity =
In symbols, it is equivalent to:
= A2 – A1 / A1 = e / A1
A1 = A2 – A1
A2 = A1 + A1
A2 = A1 (1 + )
Increase in Area = A2 – A1
Where = Superficial expansivity
A2 = Area of metal at temperature (1)
A1 = Area of metal at temperature (2)
T = temperature rise which is given by (T2 – T1)
e = 2 –A1 = expansion or increase in Area
Also the Area expansivity of solid = 2
Volume (cubic) expansivity
The Volume or Cubic expansivity is the increase in volume per unit volume per degree rise in temperature relative to that of the containing vessel or is the fractional increase in volume per Kelvin rise in temperature.
Volume Expansivity =
In symbols, it is equivalent to:
= (V2 – V1 ) / V1
Or = 3
Expansion of liquid
When a liquid is heated to a very high temperature, the molecules of the water will be vibrating about their position.
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 the container. The cubic expansivity is sometimes called the real expansivity r
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
Apparent cubic expansivity of liquid is defined as the mass of the liquid expelled per unit divided by mass left or remaining when the temperature increases by 1°C. It is measured in K-1
Relationship of apparent, volume and real/cubic/true expansivity
r =a +
= 3
r =a + 3
r = Real/cubic/true expansivity
a = Apparent expansivity
= Volume expansivity
Anomalous behaviour of water
The behaviour of water between 0°C and 4°C is termed anomalous or exceptional or unusual or irregular behaviour of water.
Worked examples
(1) A square plate of side 10cm is made of metal of linear expansivity 2 x 10-5k-1. As the plate is heated from 30°C to 100°C, the area of one face of the plate will increase to:
Solution
Area expansivity,
= 2 x 10-5k-1
(2 x 10-5k-1)
4 x 10-5k-1
Change in area, = A1 = (100)( 4 x 10-5k-1)(70)
= 0.28 cm2
New area = 100 + 0.28 = 100.28 cm2
(2) A metal cube of the cross-sectional area of 3.45m2 at 0°C is heated at a temperature rise of 70k with a length of 3m. Find the coefficient of superficial expansivity.
Solution
= (A2 – A1 )/ A1
A1 = 3.45m2 1 = 0°C
A2 = (L2)2 = (3)2 = 9m2
T= T2 – T1 = 70 – 0 = 70k
= 9 – 3.45 / 3.45 x 70
= 5.55 / 241.5
= 2.298 x 10-2k-1
In our next class, we will be talking about Heat Transfer. We hope you enjoyed the class.
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