Thermal expansion: Coefficient of superficial expansion

Coefficient of superficial expansion

The amount by which unit area of a material increases when the temperature is raised by one degree is called the coefficient of superficial (i.e. area) expansion and is represented by β (Greek beta).

If a material having an initial surface area A1 at temperature t1 and having a coefficient of superficial expansion β, has its temperature increased to t2, then the new surface area A2 of the material is given by:

New surface area = original surface area + increase in area

It is shown in Problem 5 below that the coefficient of superficial expansion is twice the coefficient of linear expansion, i.e. β = 2α, to a very close approximation.

Problem 5. Show that for a rectangular area of material having dimensions L by b, the coefficient of superficial expansion β ≈ 2α, where α is the coefficient of linear expansion.

Initial area, A1 = Lb. For a temperature rise of 1 K, side L will expand to (L + Lα) and side b will expand to (b + bα). Hence the new area of the rectangle, A2, is given by:

Coefficient of cubic expansion

The amount by which unit volume of a material increases for a one degree rise of temperature is called the coefficient of cubic (or volumetric) expansion and is represented by γ (Greek gamma).

If a material having an initial volume V1 at temperature t1 and having a coefficient of cubic expansion γ, has its temperature raised to t2, then the new volume V2 of the material is given by:

New volume = initial volume + increase in volume

It is shown in Problem 6 below that the coefficient of cubic expansion is three times the coefficient of linear expansion, i.e. γ = 3α, to a very close approximation. A liquid has no definite shape and only its cubic or volumetric expansion need be considered. Thus with expansions in liquids, equation (21.3) is used.

Problem 6. Show that for a rectangular block of material having dimensions L, b and h, the coefficient of cubic expansion γ ≈ 3α, where α is the coefficient of linear expansion.

Initial volume, V1 = Lbh. For a temperature rise of 1 K, side L expands to (L + Lα), side b expands to (b + bα) and side h expands to (h + hα)

Some typical values for the coefficient of cubic expansion measured at 20°C (i.e. 293 K) include:

The coefficient of cubic expansion γ is only constant over a limited range of temperature.

Problem 7. A brass sphere has a diameter of 50 mm at a temperature of 289 K. If the temperature of the sphere is raised to 789 K, determine the increase in (a) the diameter (b) the surface area (c) the volume of the sphere. Assume the coefficient of linear expansion for brass is 18 × 10–6 K–1.

Problem 8. Mercury contained in a thermometer has a volume of 476 mm3 at 15°C. Determine the temperature at which the volume of mercury is 478 mm3, assuming the coefficient of cubic expansion for mercury to be 1.8 × 10–4 K–1.

Initial volume, V1 = 476 mm3, final volume, V2 = 478 mm3, initial temperature, t1 = 15°C and γ = 1.8 × 10–4 K–1

Hence, the temperature at which the volume of mercury is 478 mm3 is 38.34°C

Problem 9. A rectangular glass block has a length of 100 mm, width 50 mm and depth 20 mm at 293 K. When heated to 353 K its length increases by 0.054 mm. What is the coefficient of linear expansion of the glass? Find also (a) the increase in surface area (b) the change in volume resulting from the change of length.

Now try the following Practise Exercises

Practise Exercise 113 Further problems on the coefficients of superficial and cubic expansion

1. A silver plate has an area of 800 mm2 at 15°C. Determine the increase in the area of the plate when the temperature is raised to 100°C. Assume the coefficient of linear expansion of silver to be 19 × 10–6 K–1.

[2.584 mm2]

2. At 283 K a thermometer contains 440 mm3 of alcohol. Determine the temperature at which the volume is 480 mm assuming that the coefficient of cubic expansion of the alcohol is 12 × 10–4 K–1. [358.8 K]

3. A zinc sphere has a radius of 30.0 mm at a temperature of 20°C. If the temperature of the sphere is raised to 420°C, determine the increase in: (a) the radius, (b) the surface area, (c) the volume of the sphere. Assume the coefficient of linear expansion for zinc to be 31 × 10–6 K–1.

[(a) 0.372 mm (b) 280.5 mm2 (c) 4207 mm3]

4. A block of cast iron has dimensions of 50 mm by 30 mm by 10 mm at 15°C. Determine the increase in volume when the temperature of the block is raised to 75°C. Assume the coefficient of linear expansion of cast iron to be 11 × 10–6 K–1. [29.7 mm3]

5. Two litres of water, initially at 20°C, is heated to 40°C. Determine the volume of water at 40°C if the coefficient of volumetric expansion of water within this range is 30 × 10–5 K–1. [2.012 litres]

6. Determine the increase in volume, in litres, of 3 m3 of water when heated from 293 K to boiling point if the coefficient of cubic expansion is 2.1 × 10–4 K–1 (1 litre ≈ 10–3 m3).

[50.4 litres]

7. Determine the reduction in volume when the temperature of 0.5 litre of ethyl alcohol is reduced from 40°C to –15°C. Take the coefficient of cubic expansion for ethyl alcohol as 1.1 × 10–3 K–1. [0.03025 litres]

Practise Exercise 114 Short-answer questions on thermal expansion

1. When heat is applied to most solids and liquids …….. occurs.

2. When solids and liquids are cooled they usually ……..

3. State three practical applications where the expansion of metals must be allowed for.

4. State a practical disadvantage where the expansion of metals occurs.

5. State one practical advantage of the expansion of liquids.

6. What is meant by the ‘coefficient of expansion’.

7. State the symbol and the unit used for the coefficient of linear expansion.

8. Define the ‘coefficient of superficial expansion’ and state its symbol.

9. Describe how water displays an unexpected effect between 0°C and 4°C.

10. Define the ‘coefficient of cubic expansion’ and state its symbol.

Practise Exercise 115 Multiple-choice questions on thermal expansion

1. When the temperature of a rod of copper is increased, its length:

(a) stays the same

(b) increases

(c) decreases

2. The amount by which unit length of a ma- terial increases when the temperature is raised one degree is called the coefficient of:

(a) cubic expansion

(b) superficial expansion

(c) linear expansion

3. The symbol used for volumetric expansion is:

(a) γ

(b) β

(c) L

(d) α

4. A material of length L1 at temperature θ1 K is subjected to a temperature rise of θ K. The coefficient of linear expansion of the material is α K–1. The material expands by:

(a) L2 (1 + αθ)

(b) L1 α(θ – θ1)

(c) L1 [1 + α(θ – θ1)]

(d) L1 αθ

5. Some iron has a coefficient of linear expan- sion of 12 × 10–6 K–1. A 100 mm length of iron piping is heated through 20 K. The pipe extends by:

(a) 0.24 mm

(b) 0.024 mm

(c) 2.4 mm

(d) 0.0024 mm

6. If the coefficient of linear expansion is A, the coefficient of superficial expansion is B and the coefficient of cubic expansion is C, which of the following is false?

(a) C = 3A

(b) A = B/2

(c) B = 3 C

(d) A = C/3 2

7. The length of a 100 mm bar of metal in- creases by 0.3 mm when subjected to a temperature rise of 100 K. The coefficient of linear expansion of the metal is:

(a) 3 × 10–3 K–1

(b) 3 × 10–4 K–1

(c) 3 × 10–5 K–1

(d) 3 × 10–6 K–1

8. A liquid has a volume V1 at temperature θ1.

The temperature is increased to θ2. If γ is the coefficient of cubic expansion, the increase in volume is given by:

(a) V1 γ(θ2 – θ1)

(b) V1 γ θ2

(c) V1 + V1 γ θ2

(d) V1 [1 + γ(θ2 – θ1)]

9. Which of the following statements is false?

(a) Gaps need to be left in lengths of rail- way lines to prevent buckling in hot weather.

(b) Bimetallic strips are used in thermo- stats, a thermostat being a temperature- operated switch.

(c) As the temperature of water is de- creased from 4°C to 0°C contraction occurs.

(d) A change of temperature of 15°C is equivalent to a change of temperature of 15 K

10. The volume of a rectangular block of iron at a temperature t1 is V1. The temperature is raised to t2 and the volume increases to V2. If the coefficient of linear expansion of iron is α, then volume V1 is given by: