Heat energy and transfer: Specific heat capacity.

Specific heat capacity

The specific heat capacity of a substance is the quantity of heat energy required to raise the temperature of 1 kg of the substance by 1°C. The symbol used for specific heat capacity is c and the units are J/(kg °C) or J/(kg K). (Note that these units may also be written as J kg–1 °C–1 or J kg–1 K–1).

Some typical values of specific heat capacity for the range of temperature 0°C to 100°C include:

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Hence to raise the temperature of 1 kg of iron by 1°C requires 500 J of energy, to raise the temperature of 5 kg of iron by 1°C requires (500 × 5) J of energy, and to raise the temperature of 5 kg of iron by 40°C re- quires (500 × 5 × 40) J of energy, i.e. 100 kJ.

In general, the quantity of heat energy, Q, required to raise a mass m kg of a substance with a specific heat capacity of c J/(kg °C), from temperature t1 °C to t2 °C is given by:

Q = mc(t2 t1) joules

Problem 3. Calculate the quantity of heat required to raise the temperature of 5 kg of water from 0°C to 100°C. Assume the specific heat capacity of water is 4200 J/(kg °C).

Quantity of heat energy,

Q = mc(t2 – t1)

= 5 kg × 4200 J/(kg °C) × (100 – 0)°C

= 5 × 4200 × 100

= 2100000 J or 2100 kJ or 2.1 MJ

Problem 4. A block of cast iron having a mass of  10 kg cools from a temperature of 150°C to 50°C.

How much energy is lost by the cast iron? Assume the specific heat capacity of iron is 500 J/(kg °C).

Quantity of heat energy,

Q = mc(t2 – t1)

= 10 kg × 500 J/(kg °C) × (50 – 150)°C

= 10 × 500 × (–100)

= – 500000 J or – 500 kJ or – 0.5 MJ

(Note that the minus sign indicates that heat is given out or lost).

Problem 5. Some lead having a specific heat capacity of 130 J/(kg °C) is heated from 27°C to its melting point at 327°C. If the quantity of heat required is 780 kJ, determine the mass of the lead.

Quantity of heat, Q = mc(t2 – t1), hence,

780 × 103 J = m × 130 J/(kg °C) × (327 – 27)°C i.e. 780000 = m × 130 × 300

from which, mass, m = 780000/130*300 kg = 20 kg

Problem 6. 273 kJ of heat energy are required to raise the temperature of 10 kg of copper from 15°C to 85°C. Determine the specific heat capacity of copper.

Quantity of heat, Q = mc(t2 – t1), hence:
273 × 103 J = 10 kg × c × (85 – 15)°C
where c is the specific heat capacity,
i.e. 273000 = 10 × c × 70
from which, specific heat capacity, c =
273000/10 × 70
= 390 J/(kg °C

Problem 7. 5.7 MJ of heat energy are supplied to 30 kg of aluminium that is initially at a temperature of 20°C. If the specific heat capacity of aluminium is 950 J(kg°C), determine its final temperature.

of 20°C. If the specific heat capacity of aluminium is 950 J(kg°C), determine its final temperature.

Quantity of heat,

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Problem 8. A copper container of mass 500 g contains 1 litre of water at 293 K. Calculate the quantity of heat required to raise the temperature of the water and container to boiling point, assuming there are no heat losses. Assume that the specific heat capacity of copper is 390 J/(kg K), the specific heat capacity of water is 4.2 kJ(kg K) and 1 litre of water has a mass of 1 kg.

Heat is required to raise the temperature of the water, and also to raise the temperature of the copper container.

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Practise Exercise 107 Further problems on specific heat capacity

1. Determine the quantity of heat energy (in megajoules) required to raise the tem- perature of 10 kg of water from 0°C to 50°C. Assume the specific heat capacity of water is 4200 J/(kg °C). [2.1 MJ]

2. Some copper, having a mass of 20 kg, cools from a temperature of 120°C to 70°C. If the specific heat capacity of copper is 390 J/(kg °C), how much heat energy is lost by the copper ? [390 kJ]

3. A block of aluminium having a specific heat capacity of 950 J/(kg °C) is heated from 60°C to its melting point at 660°C. If the quantity of heat required is 2.85 MJ, de- termine the mass of the aluminium block. [5 kg]

4. 20.8 kJ of heat energy is required to raise the temperature of 2 kg of lead from 16°C to 96°C. Determine the specific heat capacity of lead. [130 J/kg °C]

5. 250 kJ of heat energy is supplied to 10 kg of iron which is initially at a temperature of 15°C. If the specific heat capacity of iron is 500 J/(kg °C) determine its final tempera- ture. [65°C]

Change of state

A material may exist in any one of three states – solid, liquid or gas. If heat is supplied at a constant rate to some ice initially at, say, –30°C, its temperature rises as shown in Figure 20.1. Initially the temperature in- creases from –30°C to 0°C as shown by the line AB. It then remains constant at 0°C for the time BC required for the ice to melt into water.

When melting commences the energy gained by continual heating is offset by the energy required for the change of state and the temperature remains constant even though heating is continued. When the ice is completely melted to water, continual heating raises the temperature to 100°C, as shown by CD in Figure 20.1. The water then begins to boil and the temperature

image

again remains constant at 100°C, shown as DE, until all the water has vaporised.

Continual heating raises the temperature of the steam as shown by EF in the region where the steam is termed superheated.

Changes of state from solid to liquid or liquid to gas occur without change of temperature and such changes are reversible processes. When heat energy flows to or from a substance and causes a change of temperature, such as between A and B, between C and D and be- tween E and F in Figure 20.1, it is called sensible heat (since it can be ‘sensed’ by a thermometer).

Heat energy which flows to or from a substance while the temperature remains constant, such as be- tween B and C and between D and E in Figure 20.1, is called latent heat (latent means concealed or hidden).

Problem 9. Steam initially at a temperature of 130°C is cooled to a temperature of 20°C below the freezing point of water, the loss of heat energy being at a constant rate. Make a sketch, and briefly explain, the expected temperature/time graph rep- resenting this change.

A temperature/time graph representing the change is shown in Figure 20.2. Initially steam cools until it reaches the boiling point of water at 100°C. Tempera- ture then remains constant, i.e. between A and B, even though it is still giving off heat (i.e. latent heat). When all the steam at 100°C has changed to water at 100°C it starts to cool again until it reaches the freezing point of water at 0°C. From C to D the temperature again remains constant (i.e. latent heat), until all the water is converted to ice. The temperature of the ice then decreases as shown.

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Practise Exercise 108 A further problem on change of state

1. Some ice, initially at – 40°C, has heat sup- plied to it at a constant rate until it becomes superheated steam at 150°C. Sketch a typical temperature/time graph expected and use it to explain the difference between sensible and latent heat.

[Similar to Figure 20.1, page 228]

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