Physics and Mathematics
Heat Capacity or Thermal Capacity
1. Concept Overview
The heat capacity (also called thermal capacity) of a body is defined as the amount of heat required to raise the temperature of the entire body by 1 Kelvin (or 1°C).
It measures the ability of a body to absorb heat before its temperature increases.
In contrast to specific heat capacity (which is per unit mass), heat capacity refers to the whole body, depending on both its mass and material composition.
2. Explanation and Mathematical Derivation
Let:
- [Q] = heat absorbed by the body,
- [\Delta T] = rise in temperature,
- [C] = heat capacity of the body.
Then by definition:
[C = \dfrac{Q}{\Delta T}]
or equivalently,
[Q = C , \Delta T]
Thus, a larger heat capacity means that more heat is needed to produce the same rise in temperature.
Relation between Heat Capacity and Specific Heat Capacity
If the mass of the body is [m] and the specific heat capacity is [c], then:
[Q = m c \Delta T]
Comparing with [Q = C \Delta T], we get:
[C = m c]
Hence,
Heat Capacity = Mass × Specific Heat Capacity
This shows that heat capacity depends on the amount of substance, while specific heat is an intrinsic property.
3. Dimensions and Units
| Quantity | Symbol | SI Unit | Dimensions |
|---|---|---|---|
| Heat (energy) | [Q] | J | [M L² T⁻²] |
| Temperature Change | [\Delta T] | K | [K] |
| Heat Capacity | [C] | J·K⁻¹ | [M L² T⁻² K⁻¹] |
4. Key Features
- Heat capacity represents the thermal inertia of a body — resistance to temperature change.
- It depends on mass, specific heat, and nature of material.
- The greater the mass or specific heat, the larger the heat capacity.
- It is an extensive property, meaning it depends on the total amount of material.
- Heat capacity is independent of the path of heat transfer.
- A body with high heat capacity warms up and cools down slowly.
- The unit J·K⁻¹ indicates energy required per degree of temperature change.
- At constant pressure or volume (for gases), we define Cₚ and Cᵥ respectively.
5. Important Formulas to Remember
| Formula | Description |
|---|---|
| [C = \dfrac{Q}{\Delta T}] | Definition of heat capacity |
| [Q = C \Delta T] | Heat absorbed or released |
| [C = m c] | Relation with specific heat |
| [C_p – C_v = R] | Relation for gases (Mayer’s formula) |
6. Conceptual Questions with Solutions
1. What is meant by heat capacity?
It is the amount of heat required to raise the temperature of a body by 1 K.
2. Write the SI unit of heat capacity.
The SI unit is joule per kelvin (J·K⁻¹).
3. Is heat capacity an intensive or extensive property?
It is an extensive property because it depends on the amount of substance.
4. What is the relation between heat capacity and specific heat?
[\; C = m \, c \;], where [m] is mass and [c] is specific heat capacity.
5. What does a high heat capacity signify?
It means the substance requires a large amount of heat to raise its temperature slightly.
6. Give the dimensions of heat capacity.
[\; [M L^2 T^{-2} K^{-1}] \;]
7. What is the heat capacity of 2 kg of copper with specific heat 400 J·kg⁻¹·K⁻¹?
[\; C = m \, c = 2 \times 400 = 800 \, J·K^{-1} \;]
8. Why does water have a high heat capacity?
Because of hydrogen bonding, it requires more energy to raise its temperature.
9. What is the physical significance of heat capacity?
It measures the ability of a body to store thermal energy.
10. What is meant by thermal inertia?
It is the resistance of a body to temperature change, proportional to its heat capacity.
11. Does heat capacity depend on temperature?
Yes, slightly — it may vary with temperature, especially for gases.
12. What is the unit of heat capacity in CGS system?
cal·K⁻¹
13. What is the heat capacity of 5 kg of aluminum having c = 900 J·kg⁻¹·K⁻¹?
[\; C = m \, c = 5 \times 900 = 4500 \, J·K^{-1} \;]
14. What is the smallest possible heat capacity?
Zero — for an idealized perfectly insulated system.
15. Why does land heat up faster than water?
Because land has lower specific heat and thus lower heat capacity.
7. FAQ / Common Misconceptions
1. Is heat capacity and specific heat the same?
No. Heat capacity is for the entire body; specific heat is per unit mass.
2. Can two substances have the same specific heat but different heat capacities?
Yes, if their masses are different.
3. Does heat capacity depend on the material?
Yes, because it depends on specific heat capacity and mass.
4. If temperature doesn’t rise, does that mean heat capacity is infinite?
Not necessarily — it could mean a phase change is occurring.
5. Can heat capacity ever be negative?
No, because adding heat always increases temperature (except in exotic astrophysical systems).
6. Is heat capacity constant for all temperatures?
No, it can vary with temperature.
7. Does a large body always have a large heat capacity?
Usually yes, because heat capacity is proportional to mass.
8. Why does water act as a temperature regulator?
Because of its very high heat capacity.
9. Is heat capacity dependent on the state of matter?
Yes, it differs for solids, liquids, and gases.
10. What is the difference between heat capacity and molar heat capacity?
Heat capacity is for the whole object; molar heat capacity is per mole.
8. Practice Questions with Step-by-Step Solutions
Q1. A copper block of mass 3 kg has specific heat 400 J·kg⁻¹·K⁻¹. Find its heat capacity.
Solution:
[C = m c = 3 \times 400 = 1200 J·K^{-1}]
Heat capacity = 1200 J·K⁻¹
Q2. How much heat is required to raise the temperature of the above block by 25 K?
Solution:
[Q = C \Delta T = 1200 \times 25 = 30000 J]
Q = 30 kJ
Q3. A 0.5 kg object with specific heat 900 J·kg⁻¹·K⁻¹ is heated by 4500 J. Find the temperature rise.
Solution:
[Q = m c \Delta T \Rightarrow \Delta T] [= \dfrac{Q}{m c}] [= \dfrac{4500}{0.5 \times 900}] [= 10 K]
Temperature rise = 10 K
Q4. What will be the ratio of heat capacities of two bodies having the same material but masses 2 kg and 4 kg?
Solution:
Since [C = m c],
[C_1 : C_2 = 2 : 4 = 1 : 2]
Ratio = 1 : 2
Q5. A solid of heat capacity 100 J·K⁻¹ absorbs 500 J of heat. What is its rise in temperature?
Solution:
[\Delta T] [= \dfrac{Q}{C}] [= \dfrac{500}{100}] [= 5 K]
Temperature rise = 5 K
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