Physics and Mathematics
Effect of Temperature on Density
1. Concept Overview
The density of a substance depends on both its mass and volume.
Since mass remains constant when temperature changes, any change in volume (due to thermal expansion or contraction) results in a change in density.
Mathematically:
[\rho = \dfrac{m}{V}]
When the temperature of a substance increases, its volume usually increases (expansion), and hence, its density decreases.
Conversely, when the temperature decreases, the substance contracts, and its density increases.
2. Explanation and Mathematical Derivation
Let:
- Initial volume = [V₀]
- Final volume after heating by [ΔT] = [V]
- Initial density = [\rho₀]
- Final density = [\rho]
From the definition of volumetric expansion:
[V = V₀(1 + \gamma \Delta T)]
Since the mass [m] remains constant:
[\rho = \dfrac{m}{V}] [= \dfrac{m}{V₀(1 + \gamma \Delta T)}]
[\Rightarrow \rho] [= \dfrac{\rho₀}{(1 + \gamma \Delta T)}]
For small temperature changes, we can use the binomial approximation:
[(1 + x)^{-1} \approx 1 – x]
Hence,
[\rho \approx \rho₀(1 – \gamma \Delta T)]
This shows that density decreases linearly with rise in temperature, for small temperature changes.
3. Dimensions and Units
| Quantity | Symbol | SI Unit | Dimensions |
|---|---|---|---|
| Density | [\rho] | kg·m⁻³ | [M L⁻³] |
| Coefficient of Volumetric Expansion | [\gamma] | K⁻¹ | [K⁻¹] |
| Temperature | [T] | K | [K] |
4. Key Features
- Density is inversely proportional to the volume of a substance.
- When temperature increases, volume increases and density decreases.
- The relation [\rho = \dfrac{\rho₀}{1 + \gamma , \Delta T}] applies for solids and liquids.
- For gases, the relation is more complex (see Gas Laws).
- The rate of density change depends on the coefficient of volumetric expansion (γ).
- For small temperature changes, the decrease in density is linear.
- The principle explains phenomena like hot air rising, convection currents, and thermal stratification in fluids.
- In water, this relation shows an exception between 0°C and 4°C (explained in Anomalous Expansion of Water).
5. Important Formulas to Remember
| Formula | Description |
|---|---|
| [\rho = \dfrac{m}{V}] | Definition of density |
| [V = V₀(1 + \gamma \Delta T)] | Volumetric expansion relation |
| [\rho] [= \dfrac{\rho₀}{(1 + \gamma \Delta T)}] | Density–temperature exact relation |
| [\rho \approx \rho₀(1 – \gamma \Delta T)] | Approximate relation for small ΔT |
| [\dfrac{\Delta \rho}{\rho₀} = -\gamma \Delta T] | Fractional change in density |
6. Conceptual Questions with Solutions
1. Why does density decrease with rise in temperature?
Because when a body is heated, its volume increases while its mass remains constant, hence density decreases.
2. Does temperature affect mass?
No, heating changes only the volume, not the mass.
3. What is the mathematical relation between density and temperature?
[\rho = \dfrac{\rho₀}{1 + \gamma \, \Delta T}]
4. What happens to density when temperature decreases?
The volume decreases, so density increases.
5. How is the coefficient of volumetric expansion related to density change?
A larger γ means a larger decrease in density with temperature rise.
6. For what materials is the change in density negligible?
For solids, as their expansion (γ) is very small.
7. Why do hot air balloons rise?
Because hot air inside the balloon becomes less dense than the surrounding cool air.
8. How does density affect convection in liquids?
Hot liquid becomes less dense and rises, while cold liquid sinks, creating convection currents.
9. Does density always decrease with temperature?
Generally yes, except in special cases like water between 0°C and 4°C.
10. Why is ice less dense than water?
Because ice has a larger volume for the same mass due to its molecular structure.
11. How does thermal expansion explain floating of hot air?
Hot air expands, its volume increases, density decreases, making it buoyant.
12. If γ = 5 × 10⁻⁵ K⁻¹ and ΔT = 20°C, by what fraction does density change?
[\dfrac{\Delta \rho}{\rho₀}] [= -\gamma \Delta T] = [-5 \times 10^{-5} \times 20] [= -0.001] Hence, density decreases by 0.1%.
13. Why is γ taken as positive?
Because it measures increase in volume with rise in temperature.
14. Does γ remain constant for all temperature ranges?
No, it varies slightly with temperature and phase of the material.
15. Which physical law is used for gases in similar analysis?
The Ideal Gas Law: [PV = nRT], which links pressure, volume, and temperature.
7. FAQ / Common Misconceptions
1. Does heating always reduce density?
Not always — for water between 0°C and 4°C, density increases with temperature.
2. Does temperature change mass?
No, temperature affects volume, not mass.
3. Are solids affected the same as gases?
No, gases have much higher expansion, hence greater change in density.
4. Does pressure influence this relation?
Yes, at constant pressure, expansion causes density decrease.
5. Is γ the same for all materials?
No, each material has its own coefficient of volumetric expansion.
6. Is the relation exact for large ΔT?
No, it’s an approximation valid for small temperature differences.
7. Why is the density of air near the ceiling lower?
Because warm air rises due to lower density caused by higher temperature.
8. Does the density of liquids change significantly with temperature?
Yes, though less than gases, more than solids.
9. Can expansion cause structural stress?
Yes, in solids with constrained expansion, thermal stress may develop.
10. Why is this concept important in physics?
It explains natural phenomena like convection, buoyancy, and atmospheric circulation.
8. Practice Questions with Step-by-Step Solutions
Q1. The density of a liquid at 0°C is [1000 kg/m³]. Its coefficient of volumetric expansion is [5 \times 10^{-4} K^{-1}]. Find its density at 40°C.
Solution:
[\rho] [= \dfrac{\rho₀}{1 + \gamma \Delta T}]
[\rho] [= \dfrac{1000}{1 + 5 \times 10^{-4} \times 40}] [= \dfrac{1000}{1.02}] [= 980.4 kg/m³]
Density decreases to 980.4 kg/m³.
Q2. If [\rho₀ = 800 kg/m³] and [\gamma = 6 \times 10^{-4} K^{-1}], find the percentage change in density when the temperature rises by 30°C.
Solution:
[\dfrac{\Delta \rho}{\rho₀}] [= -\gamma \Delta T] [= -6 \times 10^{-4} \times 30 = -0.018]
Percentage change = 1.8% decrease in density.
Q3. A solid has a volumetric expansion coefficient of [4 \times 10^{-5} K^{-1}]. By how much will its density change when heated through 100°C?
Solution:
[\dfrac{\Delta \rho}{\rho₀}] [= -\gamma \Delta T = -4 \times 10^{-5} \times 100] [= -0.004]
So, density decreases by 0.4%.
Q4. Show that for small temperature changes, [\dfrac{\Delta \rho}{\rho₀}] [= -\gamma \Delta T].
Solution:
From [\rho] [= \dfrac{\rho₀}{1 + \gamma \Delta T}],
Apply binomial expansion:
[\rho \approx \rho₀(1 – \gamma \Delta T)]
Hence,
[\dfrac{\Delta \rho}{\rho₀}] [= -\gamma \Delta T]
Q5. A liquid has density [900 , kg/m³] at 20°C and [\gamma = 4 \times 10^{-4} K^{-1}]. Find its density at 70°C.
Solution:
[\rho] [= \dfrac{900}{1 + 4 \times 10^{-4} \times (70 – 20)}] [= \dfrac{900}{1.02}] [= 882.35 kg/m³]
Density decreases due to thermal expansion.
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