The coefficient of areal expansion of a material is 1·6 × 10⁻⁵ K⁻¹. Which one of the following gives the value of coefficient of volume expansion of this material?
- (a)0·8 × 10⁻⁵ K⁻¹
- (b)2·4 × 10⁻⁵ K⁻¹
- (c)3·2 × 10⁻⁵ K⁻¹
- (d)4·8 × 10⁻⁵ K⁻¹
Correct — B, 2·4 × 10⁻⁵ K⁻¹. For an isotropic solid the three coefficients of thermal expansion stand in the fixed ratio α : β : γ = 1 : 2 : 3, where α is linear, β areal and γ volumetric. The reason is straightforward, because a length grows by a factor (1 + αΔT), so an area, being the product of two lengths, grows by roughly (1 + 2αΔT) and a volume, the product of three, by roughly (1 + 3αΔT). Here the areal coefficient is given, so first step back to the linear one, α = β/2 = 1·6 × 10⁻⁵ / 2 = 0·8 × 10⁻⁵ K⁻¹, and then forward to the volumetric one, γ = 3α = 2·4 × 10⁻⁵ K⁻¹. Equivalently, γ = 1·5 β in a single move.
- (a)0·8 × 10⁻⁵ K⁻¹ — This is the linear coefficient α, obtained by halving the given areal value. It is a genuine step on the way to the answer, but the question asks for the volumetric coefficient, which is three times this, not one times.
- (c)3·2 × 10⁻⁵ K⁻¹ — This is twice the given areal coefficient. Doubling is the step that takes you from linear to areal, not from areal to volumetric, so the operation has been applied at the wrong stage.
- (d)4·8 × 10⁻⁵ K⁻¹ — This is three times the areal coefficient. The factor of three belongs between the linear and the volumetric coefficients; applying it to β overshoots by a factor of two.
Heating a solid increases the average separation between its atoms, so every dimension grows. The fractional change per degree is the coefficient of expansion, and it takes three forms — linear for a length, areal or superficial for a surface, and cubical or volumetric for a volume. In an isotropic material the same underlying stretch shows up once in a line, twice in an area and three times in a volume, giving the ratio 1 : 2 : 3.
This is a ratio question wearing the costume of a numerical one, and the only real decision is which multiple to apply. The four options are deliberately built as ×0·5, ×1·5, ×2 and ×3 of the given figure, so a candidate who remembers only 'there is a factor of two and a factor of three somewhere' has a one-in-four chance. Fix the chain instead, running from linear to areal by doubling and then on to volumetric by a further one and a half. It also helps to remember that these coefficients are small — of the order of 10⁻⁵ per kelvin for common solids — which is why bridges need expansion joints and railway tracks are laid with gaps, yet a metal rod does not visibly lengthen in the sun.
- For an isotropic solid, α : β : γ = 1 : 2 : 3, where α is linear, β areal and γ volumetric.
- From areal to volumetric directly, γ = 1·5 β; here 1·5 × 1·6 × 10⁻⁵ = 2·4 × 10⁻⁵ K⁻¹.
- The coefficients are reciprocal temperature, so the units are per kelvin, and a change of 1 K equals a change of 1 °C.
- Liquids and gases have no linear or areal coefficient of their own — only a coefficient of volume expansion is defined for them.
Halve to reach the linear coefficient, then triple it — a net factor of one and a half.
- Multiplying the areal coefficient by three instead of by one and a half.
- Stopping at the linear coefficient, which is one of the options and looks like a completed calculation.
- Applying the 1 : 2 : 3 ratio to an anisotropic crystal, where the linear coefficient differs along different axes.
Asked as a one-step conversion between expansion coefficients, with every plausible wrong multiple offered as an option.
Why is it difficult to measure the coefficient of expansion of a liquid than solid ?
- (a) Liquids tend to evaporate at all temperatures
- (b) Liquids conduct more heat
- (c) Liquids expand too much when heated
- (d) Their containers also expand when heated
Answer(d) Their containers also expand when heated
Extends the same coefficient into liquids, where only volume expansion exists and the vessel's own expansion has to be subtracted — the distinction between apparent and real expansion.
- practice — not a real PYQ
The coefficient of linear expansion of a metal is 1·2 × 10⁻⁵ K⁻¹. Its coefficient of volume expansion is
- (a)0·4 × 10⁻⁵ K⁻¹
- (b)2·4 × 10⁻⁵ K⁻¹
- (c)3·6 × 10⁻⁵ K⁻¹
- (d)1·2 × 10⁻⁵ K⁻¹
Answer(c) 3·6 × 10⁻⁵ K⁻¹ — the volumetric coefficient is three times the linear one for an isotropic solid.
- practice — not a real PYQ
For a liquid, which of the following coefficients of expansion is normally defined?
- (a)Linear only
- (b)Areal only
- (c)Volumetric only
- (d)All three equally
Answer(c) Volumetric only — a liquid has no fixed shape, so it has no length or surface of its own to expand; only its volume can be tracked.