Physics and Mathematics
Water Equivalent
1. Concept Overview
The water equivalent of a body is defined as the mass of water that would absorb or release the same amount of heat as the body for the same change in temperature.
It gives an equivalent measure of how a material behaves thermally compared to water.
Since water has a known specific heat capacity, we can easily compare any substance’s heat-absorbing ability by finding its water equivalent.
2. Explanation and Mathematical Derivation
Let
- [m] = mass of the body,
- [c] = specific heat capacity of the material,
- [W] = water equivalent of the body,
- [c_w] = specific heat capacity of water,
- [\Delta T] = rise in temperature.
Now, heat absorbed by the body:
[
Q = m c \Delta T
]
Heat absorbed by water equivalent mass of water:
[
Q = W c_w \Delta T
]
Since both absorb equal amounts of heat for the same temperature change,
[m c \Delta T = W c_w \Delta T]
Cancelling [\Delta T]:
[
\boxed{W = \dfrac{m c}{c_w}}
]
Hence,
Water Equivalent (W) is the mass of water having the same heat capacity as that of the body.
3. Dimensions and Units
| Quantity | Symbol | SI Unit | Dimensions |
|---|---|---|---|
| Water Equivalent | [W] | kg | [M] |
(It represents a mass, not energy.)
4. Key Features
- Water equivalent gives a simple way to express heat capacity in terms of water mass.
- It depends on both mass (m) and specific heat capacity (c) of the body.
- For water itself, [W = m], since [c = c_w].
- It helps in calorimetry experiments, where water is used as a reference medium.
- If two bodies have the same water equivalent, they absorb equal heat for the same rise in temperature.
- It is a measure of heat capacity, but expressed in mass units (kg or g).
- It is used to correct for the thermal capacity of the calorimeter.
- The larger the water equivalent, the greater the ability of the body to absorb heat.
5. Important Formulas to Remember
| Formula | Description |
|---|---|
| [W = \dfrac{m c}{c_w}] | Definition of water equivalent |
| [C = W c_w] | Heat capacity in terms of water equivalent |
| [Q = W c_w \Delta T] | Heat absorbed using water equivalent |
| [W = m] | For water itself |
6. Conceptual Questions with Solutions
1. Define water equivalent.
It is the mass of water that absorbs or releases the same amount of heat as the body for the same temperature change.
2. Write the formula for water equivalent.
[\; W = \dfrac{m \, c}{c_w} \;]
3. What is the SI unit of water equivalent?
Kilogram (kg)
4. What is the dimension of water equivalent?
[\; [M] \;]
5. What is the water equivalent of 1 kg of water?
[\; W = m = 1 \, \text{kg} \;]
6. What does a large value of water equivalent signify?
It means the body can absorb more heat for the same temperature rise.
7. A body has m = 2 kg, c = 500 J·kg⁻¹·K⁻¹, cₜ = 4200 J·kg⁻¹·K⁻¹. Find its water equivalent.
[\; W = \dfrac{2 \times 500}{4200} = 0.238 \, \text{kg} \;]
8. What is the physical meaning of water equivalent?
It represents how much water would behave thermally like the given substance for heat absorption.
9. Is water equivalent an extensive or intensive property?
It is an extensive property, as it depends on the amount of material.
10. What is the water equivalent of a copper block with m = 200 g, c = 0.1 cal·g⁻¹·°C⁻¹?
[\; W = \dfrac{200 \times 0.1}{1} = 20 \, \text{g} \;]
11. In calorimetry, why do we calculate water equivalent of a calorimeter?
To account for the heat absorbed by the calorimeter material itself, so results are more accurate.
12. If specific heat of copper = 0.39 J·g⁻¹·K⁻¹ and that of water = 4.18 J·g⁻¹·K⁻¹, find W for 500 g copper.
[\; W = \dfrac{500 \times 0.39}{4.18} = 46.7 \, \text{g} \;]
13. What happens to W if specific heat of the body increases?
W increases, since [W ∝ c].
14. If c = c_w, what will be W?
[\; W = m \;]
15. Why is water chosen as a reference for heat measurements?
Because its specific heat is high and precisely known.
7. FAQ / Common Misconceptions
1. Is water equivalent measured in joules?
No, it is measured in kilograms or grams, as it represents a mass.
2. Is water equivalent the same as heat capacity?
They are related but not the same. [C = W \, c_w].
3. Can two bodies have the same heat capacity but different water equivalents?
No, since [W = \dfrac{C}{c_w}], equal C means equal W.
4. Does water equivalent depend on temperature?
Slightly, because specific heat can change with temperature.
5. If W = 100 g, what does it mean physically?
It means the body behaves thermally like 100 g of water.
6. Is W smaller for metals compared to water?
Yes, because metals have much smaller specific heats.
7. Is W always less than m?
Yes, for substances with [c < c_w].
8. Can W ever be greater than m?
Only if [c > c_w], which is rare.
9. Why is the concept of W useful in calorimetry?
It allows replacement of body’s heat capacity with an equivalent water mass, simplifying calculations.
10. What is the relation between water equivalent and thermal capacity?
[\; C = W \, c_w \;]
8. Practice Questions with Step-by-Step Solutions
Q1. Find the water equivalent of a 200 g copper vessel (specific heat 0.39 J·g⁻¹·K⁻¹).
Given: [c_w = 4.18 J·g⁻¹·K⁻¹]
Solution:
[W = \dfrac{200 \times 0.39}{4.18}] [= 18.7 \text{g}]
Water Equivalent = 18.7 g
Q2. A brass body has water equivalent 50 g. Find its heat capacity.
Solution:
[C = W c_w = 50 \times 4.18] [= 209 J·K^{-1}]
C = 209 J·K⁻¹
Q3. A 500 g aluminum block has specific heat 0.9 J·g⁻¹·K⁻¹. Find its water equivalent.
Solution:
[W = \dfrac{500 \times 0.9}{4.18}] [= 107.7 \text{g}]
W = 107.7 g
Q4. A body has heat capacity 800 J·K⁻¹. Find its water equivalent.
Solution:
[W = \dfrac{C}{c_w}] [= \dfrac{800}{4200}] [= 0.190 \text{kg}]
W = 0.19 kg
Q5. A copper calorimeter has water equivalent 45 g. If its temperature rises by 10°C, how much heat does it absorb?
Solution:
[Q = W c_w \Delta T] [= 45 \times 4.18 \times 10] [= 1881 \text{J}]
Q = 1.88 kJ
Sign in with Google