# Physics: Thermal, linear, area and cubic expansivity

## Expansivity

This is defined as amount a material expands or contracts per unit length, per-degree change in temperature.
The addition of heat will usually cause expansion of the body. during expansion, the dimension of the body increases.

### Areas covered

• concept of thermal expansivity
• Linear Expansion
• Linear Expansivity
• Area expansivity
• Cubic or volume expansivity
• Questions and solutions

### 1) Concept of thermal expansivity

When heated most solid and liquid expand. they also contract when cooled. Expansion means an increase in size of object. According to kinetic molecular theory, when an object is heated the molecule acquire more kinetic energy which enable them to overcome intermolecular forces. Therefore, the vibration of the molecule increases and their displacement mean position increases. As a result of this, the average distance between the molecules of the substance leading to an increase in size of the substance.

#### 2) Linear Expansion

Different solid expands by different amount when heated over the same temperature range. Copper for example expands more than steel, when both are heated through the same rise in temperature. This is because they have different coefficient of linear expansion.

#### 3) Linear Expansivity  (α)

The linear expansivity of a substance is defined as an increase in length, per-unit length, per degree rise in temperature. In symbols.
This is equivalent to;

Where, α = linear expansivity, L1 = initial length, L2 = final length, ɵ2 = final temperature, ɵ1 = initial temperature.
L2 – L1 = expansion

L2 = L1 (αɵ1 + 1)
The unit of linear expansivity is per-degree or per-kelvin.

### 4) Area expansivity

Area expansivity is the increase in area, per unit area, per degree rise in temperature for a plate of initial area (AI), which expands to the final area (Af) upon a temperature rise ∆ɵ

β = 2α
therefore, A= AI (2αɵ + 1)

### 5) Cubic expansivity (ɤ)

Cubic or volume expansivity of a solid, liquid or gas is the increase in volume, per-unit volume, per-degree rise in temperature.

ɤ = 3 α
I.e 3 α =
3 α VIɵ =   VF – VI
VF = VI (3αɵ + 1)

#### Questions and solutions

1)   A copper rod whose length at 30°c is 10m is heated to 50°c. Find the new length. (Take α as 0.000017 K-1)

Solution
Recall for linear expansivity, L2 = L1 (αɵ1 + 1)
ɵ = 50°c - 30°c = 20°c
L2 = 10(0.000017(20) + 1) = 10.0034m

2) Explain the statement that the linear expansivity of brass is 0.000019 k-1. A brass rod 90cm long at 28°c is heated to 98°c. Find the change in length of a rod.

solution
L2 = L1 (αɵ1 + 1)
ɵ = 98°c - 28°c = 70°c
L2 = 90(0.000019(70) + 1) = 90.1197cm

3)  Describe an experiment to measure the linear expansivity of copper in the form of rod. Calculate the expansion of 10m of copper rod when heated from 10°c to 80°c . Take the linear expansivity of copper as 0.000017.

Solution
L2 = L1 (αɵ1 + 1)
ɵ = 80°c - 10°c = 70°c
L2 = 10(0.000017(70) + 1) = 10.0119

4)   An iron rod is 1.58m long at 0°c. What must be the length of a brass rod at 0°c if the the difference between the lengths of the two rods is to remain the same at all temperatures.

Solution
Brass rod = L1
Iron rod = 1.58m
α (brass rod) = 0.000019 K-1
α (iron rod) = 0.000012 K-1
Linear expansivity difference = 0.000007 k-1
L1 = 1.0m