# Volume Expansion

## Substances expand or contract when their temperature changes, with expansion or contraction occurring in all directions.

#### Key Points

• Substances that expand at the same rate in every direction are called isotropic.

• In the case of a gas, expansion depends on how the pressure changed in the process because the volume of a gas will vary appreciably with pressure as well as temperature.

• For a solid, we can ignore the effects of pressure on the material, and the volumetric thermal expansion coefficient can be written as $\alpha_V = \frac{1}{V} \frac{dV}{dT}$. For isotropic materials, $\alpha_V =3 \alpha_L$.

#### Terms

• Having properties that are identical in all directions; exhibiting isotropy.

• The fractional change in length per degree of temperature change.

#### Figures

1. ##### Volumetric Expansion

In general, objects expand in all directions as temperature increases. In these drawings, the original boundaries of the objects are shown with solid lines, and the expanded boundaries with dashed lines. (a) Area increases because both length and width increase. The area of a circular plug also increases. (b) If the plug is removed, the hole it leaves becomes larger with increasing temperature, just as if the expanding plug were still in place. (c) Volume also increases, because all three dimensions increase.

2. ##### Thermal Expansion - Volume Expansion

A brief introduction to thermal expansion for students.

The volumetric thermal expansion coefficient Figure 2 is the most basic thermal expansion coefficient. Figure 1 illustrates that, in general, substances expand or contract when their temperature changes, with expansion or contraction occurring in all directions. Such substances that expand in all directions are called isotropic. For isotropic materials, the area and linear coefficients may be calculated from the volumetric coefficient (discussed below).

Mathematical definitions of these coefficients are defined below for solids, liquids, and gasses:

$\alpha_V= \frac{1}{V} (\frac{\partial V}{\partial T})_p$.

The subscript p indicates that the pressure is held constant during the expansion. In the case of a gas, the fact that the pressure is held constant is important, as the volume of a gas will vary appreciably with pressure as well as with temperature.

For a solid, we can ignore the effects of pressure on the material, thus the volumetric thermal expansion coefficient can be written:

$\alpha_V = \frac{1}{V} \frac{dV}{dT}$

where V is the volume of the material, and is dV/dT the rate of change of that volume with temperature. This means that the volume of a material changes by some fixed fractional amount. For example, a steel block with a volume of 1 cubic meter might expand to 1.002 cubic meters when the temperature is raised by 50 °C. This is an expansion of 0.2%. The volumetric expansion coefficient would be 0.2% for 50 °C, or 0.004% per degree C.

### Relationship to Linear Thermal Expansion Coefficient

For isotropic material, and for small expansions, the linear thermal expansion coefficient is one third the volumetric coefficient. To derive the relationship, let's take a cube of steel that has sides of length L. The original volume will be V = L3,and the new volume, after a temperature increase, will be:

\begin{align} V+ \Delta V &= (L + \Delta L)^3 \\ &= L^3 + 3L^2\Delta L + 3L(\Delta L )^2 +(\Delta L)^3 \\ &\approx L^3 + 3L^2\Delta L \\ &= V + 3 V \frac {\Delta L}{L} \end{align}.

The approximation holds for a sufficiently small $\Delta L$ compared to L. Since:

$\frac{\Delta V}{V} = 3\frac{\Delta L}{L}$

(and from the definitions of the thermal coefficients), we arrive at:

$\alpha_V =3 \alpha_L$.

#### Key Term Glossary

approximation
An imprecise solution or result that is adequate for a defined purpose.
##### Appears in these related concepts:
contraction
A reversible reduction in size.
##### Appears in these related concepts:
gas
Matter in a state intermediate between liquid and plasma that can be contained only if it is fully surrounded by a solid (or held together by gravitational pull); it can condense into a liquid, or can (rarely) become a solid directly.
##### Appears in these related concepts:
isotropic
Having properties that are identical in all directions; exhibiting isotropy.
##### Appears in these related concepts:
liquid
A substance that is flowing, and keeping no shape, such as water; a substance of which the molecules, while not tending to separate from one another like those of a gas, readily change their relative position, and which therefore retains no definite shape, except that determined by the containing receptacle; an inelastic fluid.
##### Appears in these related concepts:
pressure
the amount of force that is applied over a given area divided by the size of that area
##### Appears in these related concepts:
solid
A substance in the fundamental state of matter that retains its size and shape without need of a container (as opposed to a liquid or gas).