Physics and Mathematics
Specific Heat Capacity or Specific Heat
1. Concept Overview
Every substance requires a certain amount of heat energy to raise its temperature.
The quantity of heat required to raise the temperature of 1 kilogram of a substance by 1 Kelvin (or 1°C) is called its Specific Heat Capacity, or simply Specific Heat.
It represents the thermal inertia of a substance — i.e., the resistance a body offers to change in its temperature when heat is supplied or removed.
2. Explanation and Mathematical Derivation
Let:
- [m] = mass of the substance
- [Q] = heat absorbed or released
- [c] = specific heat capacity
- [\Delta T] = change in temperature
Then, the heat absorbed is directly proportional to both the mass and temperature rise:
[Q \propto m , \Delta T]
By introducing the proportionality constant [c]:
[Q = m , c , \Delta T]
Therefore,
[c = \dfrac{Q}{m , \Delta T}]
This is the defining equation for specific heat capacity.
Case 1 — When heat is absorbed:
If heat is supplied, [Q] is positive, and temperature rises.
Case 2 — When heat is released:
If heat is removed, [Q] is negative, and temperature falls.
Relationship Between Heat and Temperature Rise
From the equation [Q = m c \Delta T],
- Greater the specific heat (c) → more heat required for same temperature rise.
- Smaller the specific heat (c) → faster temperature change for same amount of heat.
3. Dimensions and Units
| Quantity | Symbol | SI Unit | Dimensions |
|---|---|---|---|
| Heat Energy | [Q] | Joule (J) | [M L² T⁻²] |
| Mass | [m] | kg | [M] |
| Temperature Change | [\Delta T] | K | [K] |
| Specific Heat Capacity | [c] | J·kg⁻¹·K⁻¹ | [L² T⁻² K⁻¹] |
4. Key Features
- Specific Heat is an intrinsic property — it depends only on the nature of the substance.
- Substances with higher specific heat can store more heat without large temperature changes.
- Water has an exceptionally high specific heat, making it ideal for thermal regulation (e.g., in nature and cooling systems).
- Metals have low specific heat; they heat up and cool down quickly.
- The specific heat of a body can be measured using a calorimeter.
- Units conversion:
- 1 cal g⁻¹ °C⁻¹ = 4.186 × 10³ J kg⁻¹ K⁻¹
- For solids and liquids, specific heat is almost constant over small temperature ranges.
- For gases, specific heat depends on whether heating occurs at constant pressure (Cₚ) or constant volume (Cᵥ).
5. Important Formulas to Remember
| Formula | Description |
|---|---|
| [Q = m c \Delta T] | Heat absorbed or released |
| [c = \dfrac{Q}{m \Delta T}] | Definition of specific heat |
| [Q = m c (T₂ – T₁)] | Temperature difference form |
| [1 cal g^{-1} °C^{-1}] [= 4.186 × 10^3 J kg^{-1} K^{-1}] | Unit conversion |
| [C = m c] | Heat capacity of a body |
6. Conceptual Questions with Solutions
1. What is specific heat capacity?
It is the quantity of heat required to raise the temperature of 1 kg of a substance by 1 K.
2. What does a high specific heat signify?
It means the substance absorbs more heat for the same temperature rise — it changes temperature slowly.
3. Which has higher specific heat — water or iron?
Water; hence it heats and cools more slowly than iron.
4. Why is water used as a coolant?
Because of its high specific heat, it can absorb large quantities of heat without large temperature rise.
5. Why does sand heat up faster than water under sunlight?
Sand has a lower specific heat, so it requires less heat to increase its temperature.
6. Write the relation between Q, m, c, and ΔT.
[\; Q = m c \Delta T \;]
7. What are the units of specific heat in SI and CGS systems?
SI: J·kg⁻¹·K⁻¹ CGS: cal·g⁻¹·°C⁻¹
8. What happens to temperature when a substance with high c absorbs heat?
It rises slowly.
9. Does specific heat depend on mass?
No, it is an intensive property — independent of mass.
10. Can specific heat be negative?
No, it is always positive because heat absorption increases temperature.
11. What is the specific heat of water in SI units?
[\; 4186 \ J \ kg^{-1} \ K^{-1} \;]
12. Why does coastal climate remain moderate?
Because water’s high specific heat regulates temperature by absorbing and releasing heat slowly.
13. What is the relation between specific heat and heat capacity?
[\; C = m c \;]
14. If 100 J of heat raises the temperature of 0.5 kg of metal by 10 K, find c.
[c = \dfrac{Q}{m \Delta T}] [= \dfrac{100}{0.5 \times 10}] [= 20 \J \kg^{-1} K^{-1}]
15. Why do metals feel colder than wood at the same temperature?
Because metals conduct heat faster due to higher thermal conductivity, not lower specific heat — though both effects influence temperature perception.
7. FAQ / Common Misconceptions
1. Is specific heat the same for all materials?
No, it varies widely from one substance to another.
2. Does specific heat depend on temperature?
Slightly — it can vary with temperature and phase.
3. Is specific heat capacity an extensive property?
No, it’s intensive; it doesn’t depend on the mass of the substance.
4. Why do equal masses of different substances heat up differently?
Because their specific heats are different.
5. Is specific heat the same as heat capacity?
No, heat capacity depends on total mass, while specific heat is per unit mass.
6. Can specific heat be negative?
No, a negative specific heat would violate thermodynamic laws for ordinary materials.
7. Do gases have one specific heat value?
No, they have two — [C_p] (constant pressure) and [C_v] (constant volume).
8. Does a higher c mean a material feels hotter?
No, it means the material warms up more slowly for the same heat input.
9. Does c depend on phase (solid/liquid/gas)?
Yes, it changes when the material changes phase.
10. Is the value of c same for heating and cooling?
It may differ slightly due to heat losses and non-linearity.
8. Practice Questions with Step-by-Step Solutions
Q1. How much heat is required to raise the temperature of 2 kg of copper from 20°C to 70°C?
(Given: [c = 390 J kg^{-1} K^{-1}])
Solution:
[Q = m c \Delta T] [= 2 \times 390 \times (70 – 20)] [= 2 \times 390 \times 50 = 39{,}000 J]
Heat required = 39 kJ
Q2. What is the specific heat of a metal if 500 J of heat raises the temperature of 0.25 kg of it by 10 K?
Solution:
[c = \dfrac{Q}{m\Delta T}] [= \dfrac{500}{0.25 \times 10}] [= 200 Jkg^{-1} K^{-1}]
Q3. A 0.1 kg piece of aluminum (c = 900 J·kg⁻¹·K⁻¹) absorbs 4500 J of heat. Find its temperature rise.
Solution:
[\Delta T] [= \dfrac{Q}{mc}] [= \dfrac{4500}{0.1 \times 900} = 50 K]
Q4. A liquid requires 6300 J of heat to raise its temperature from 30°C to 60°C for 0.3 kg. Find its specific heat.
Solution:
[c = \dfrac{Q}{m \Delta T}] [= \dfrac{6300}{0.3 \times 30}] [= 700 J kg^{-1} K^{-1}]
Q5. Compare the temperature rise of equal masses of water and copper if both absorb equal heat.
Given: [c_{water} = 4186], [c_{copper} = 390].
Solution:
[\Delta T = \dfrac{Q}{m c}]
For same Q and m:
[\dfrac{\Delta T_{water}}{\Delta T_{copper}}] [= \dfrac{c_{copper}}{c_{water}}] [= \dfrac{390}{4186} \approx 0.093]
So, water’s temperature rise is only 9.3% of copper’s — water heats much more slowly.
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