Thermal Expansion Concept Page - 4

Example

Disadvantages of thermal expansion of liquid and gases

Not all materials contract/expand at the same rate; the amount of contraction/expansion varies depending on the material. The coefficient of linear expansion, is a proportionality constant that determines the rate of change in length in different materials when they are heated and cooled
The disadvantages of thermal expansion of liquids is: 
When temperature increased, the liquid gets heated and expands its volume and overflow the container. So it's difficult to keep in a closed vessel or container when the liquid is heating. We can take the example of milk, if we heat more, milk will overflow as we have seen already and this is due to thermal expansion of liquid. 
Disadvantages of thermal expansion of gases :
When gas molecules are being heated, the entropy of those gas molecules are highly increases which may cause in many ways. Let me take an example of nuclear reaction. In nuclear reaction, if the size of the gas molecules increases due to increase of temperature, there will be increase in pressure which may cause more vigorously in the environment. 

Definition

Anamolous expansion of Water

The anomalous expansion of water is an abnormal property of water whereby it expands instead of contracting when the temperature goes from 4o C$4^{o}C$ to 0o C$0^{o}C$, and it becomes less dense. The density is maximum at 4 degree centigrade and decreases below that temperature as shown in graph. The density becomes less and less as it freezes because molecules of water normally form open crystal structures when in solid form.

Diagram

Volume and density of water vs temperature

Definition

Anomalous behaviour of water using coefficient of thermal expansion

Variation of coefficient of volume expansion with temperature is shown in the attached plot. 
For 0oC<T<4oC$0^{o}C<T<4^{o}C$,
     β<0$\beta <0$
     ⟹ΔV<0$⟹\mathrm{\Delta }V<0$      
For T>4oC$T>4^{o}C$, 
     β>0$\beta >0$
     ⟹ΔV>0$⟹\mathrm{\Delta }V>0$
Hence, volume of water is minimum at a temperature of 4oC$4^{o}C$ 

Definition

Hope's experiment

Hopes apparatus consists of a long cylindrical jar with two openings on the side, one near the top and the other near the bottom to fit thermometers in each of these openings. A metallic cylindrical air-tight trough with an outlet is also fitted onto the jar, on its central portion. Now the cylindrical jar is filled with water. The cylindrical trough at the central portion of the jar is filled with a freezing mixture of ice and common salt. After a period of time the more dense water collects at the bottom at a temperature of 4C showing the anomalous behaviour of water. 

Definition

Consequences of anomalous expansion of water

Because of the anomalous expansion of water, when the temperature reaches 40C$4^{0}C$, the water remains at the surface of the lake, sea or river. This is because as temperature decreases from 40$4^0$C to 00C$0^{0}C$, water expands. The water is still less dense and thus remains at the surface. Only a small layer of water at the surface thus becomes frozen. Ice itself is less dense than water.

Result

Variation of density of water with temperature

Variation in density of water with temperature.

Example

Solving problems of density variation with temperature

The density of a liquid at 1000${100}^0$C is 8.0gm/cm3$8.0gm/c{m}^3$ and at 100${10}^0$C is 8.4gm/cm3$8.4gm/c{m}^3$. The coefficient of cubical expansion of the liquid is nearlyThe dependence of density with temperature is given as
d0=dt(1+γΔt)$d_0=d_t(1+\gamma \mathrm{\Delta }t)$ where d0$d_0$ is the reference density and dt$d_t$ is the density at temperature t.
Then,
8.4=dt(1+γ×(100−10))$8.4=d_t(1+\gamma \times (100-10))$
Here dt=8$d_t=8$ at temperature 1000C${100}^{0}C$.
∴γ=(8.48−1)90=0.0590=5.55×10−4

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