FLUID EXPANSION

Expansion of liquids

Like solids, liquids also, in general, expand on heating ; however, their expansion is much large compared to solids for the same temperature rise. A noteworthy point to be taken into account during the expansion of liquid is that they are always contained in a vessel or a container and hence the expansion of the vessel also comes into picture. Further, linear or superficial expansion in case of a liquid does not carry any sense.

Consider a liquid contained in a round bottomed flask fitted with a long narrow stem as shown in fig. Let the initial level of the liquid be \(\mathrm{X}\). When it is heated the level falls initially to \(\mathrm{Y}\).

However, after sometime, the liquid level eventually rises to \(\mathrm{Z}\). The entire phenomenon can be understood as follows: Upon being heated, the container gets heated first and hence expands. As a result, the capacity of the flask increases and hence the liquid level falls.

After sometime, the heat gets conducted from the vessel to the liquid and hence liquid also expands thereby rising its level eventually to \(\mathrm{Z}\). Since, the volume expansivity of liquids, in general, are far more than that of solids, so the level \(\mathrm{Z}\) will be above the level \(\mathrm{X}\).

Effect of temperature on density

When a solid or liquid is heated, it expands, with mass remaining constant. Density being the ratio of mass to volume, it decreases. Thus, if \(\mathrm{V}_{0}\) and \(\mathrm{V}_{\mathrm{t}}\) be the respective volumes of a substance at \(0^{\circ} \mathrm{C}\) and \(t^{\circ} \mathrm{C}\) and if the corresponding values of densities be \(\rho_{0}\) and \(\rho_{t}\), then the mass \(m\) of the substance is given by

\(\mathrm{m}=\mathrm{V}_{0} \rho_{0}=\mathrm{V}_{\mathrm{t}} \rho_{\mathrm{t}}\)

But \(\quad \mathrm{V}_{\mathrm{t}}=\mathrm{V}_{0}(1+\gamma \mathrm{t})\), so \(\rho_{\mathrm{t}}=\rho_{0}(1+\gamma \mathrm{t})^{-1} \approx \rho_{0}(1-\gamma \mathrm{t})\) (Binomial approximation) 

Illustration :

The volume of a glass vessel filled with mercury is \(500 \mathrm{cc}\), at \(25^{\circ} \mathrm{C}\). What volume of mercury will overflow at \(45^{\circ} \mathrm{C}\) ? (coefficients of volume expansion of mercury and glass are \(1.8 \times 10^{-4} /{ }^{\circ} \mathrm{C}\) and \(9.0 \times 10^{-6} /{ }^{\circ} \mathrm{C}\) respectively).

Sol. The volume of mercury overflowing will be the expansion of mercury relative to the glass vessel (i.e., apparent expansion).

Now, since \(\quad(\Delta V)_{a}=(\Delta V)_{r}-(\Delta V)_{c}\)

Apparent expansion \((\Delta V)_{a}\) will be

\( (\Delta V)_{a}=  V_{t} \gamma_{t} \Delta T-V_{c} \gamma_{c} \Delta T \)

\( =500 c c(180-9) \times \frac{10^{-6}}{{ }^{\circ} \mathrm{C}}(45-25)^{\circ} \mathrm{C} \)

 \(=1.71 c c \)

Thus, \(1.71\) cc of mercury overflows.

Anomolous expansion of water

While most substances expand when heated, a few do not. For instant, if water at \(0^{\circ} \mathrm{C}\) is heated, its volume decreases until the temperature reaches \(4^{\circ} \mathrm{C}\). Above \(4{ }^{\circ} \mathrm{C}\) water behaves normally, and its volume increases as the temperature increases.

Because a given mass of water has a minimum volume at \(4^{\circ} \mathrm{C}\), the density (mass per unit volume) of water is greatest at \(4{ }^{\circ} \mathrm{C}\), as figure shows.

The density of water in the temperature range from 0 to \(10^{\circ} \mathrm{C}\). Water has a maximum density of \(999.973 \mathrm{~kg} / \mathrm{m}^{3}\) at \(4^{\circ} \mathrm{C}\). (This value for the density is equivalent to the often quoted density of \(1.000\) grams per milliliter)

When the air temperature drops, the surface layer of water is chilled. As the temperature of the surface layer drops toward \(4^{\circ} \mathrm{C}\), this layer becomes more dense than the warmer water below. The denser water sinks and pushes up the deeper and warmer water, which in turn is chilled at the surface. This process continues until the temperature of the entire lake reaches \(4^{\circ} \mathrm{C}\). Further cooling of the surface water below \(4^{\circ} \mathrm{C}\) makes it less dense than the deeper layers ; consequently, the surface layer does not sink but stays on top. Continued cooling of the top layer to \(0^{\circ} \mathrm{C}\) leads to the formation of ice that floats on the water, because ice has a smaller density than water at any temperature. Below the ice, however, the water temperature remains above \(0^{\circ} \mathrm{C}\). The sheet of ice acts as an insulator that reduces the loss of heat from the lake, especially if the ice is covered with a blanket of snow, which is also an insulator. As a result, lakes usually do not freeze solid, even during prolonged cold spells, so fish and other aquatic life can survive.