You may have observed that sometimes sealed bottles with metallic lids are so tightly screwed that one has to put the lid in hot water for sometime to open the lid. This would allow the metallic cover to expand, thereby loosening it to unscrew easily.
In case of liquids, you may have observed that mercury in a thermometer rises, when the thermometer is put in a slightly warm water. If we take out the thermometer from the warm water the level of mercury falls again.
Similarly, in the case of gases, a balloon partially inflated in a cool room may expand to full size when placed in warm water. On the other hand, a fully inflated balloon when immersed in cold water would start shrinking due to contraction of the air inside.
It is our common experience that most substances expand on heating and contract on cooling. A change in the temperature of a body causes change in its dimensions. The increase in the dimensions of a body due to the increase in its temperature is called `"thermal expansion."`
The expansion in length is called linear expansion. The expansion in area is called area expansion. The expansion in volume is called volume expansion (Fig. 11.5).
If the substance is in the form of a long rod, then for small change in temperature, `ΔT,` the fractional change in length, `Δl//l,` is directly proportional to `ΔT.`
`color{orange}((Deltal)/l = alpha_1DeltaT)`.......(11.4)
where `α_1` is known as the coefficient of linear expansion and is characteristic of the material of the rod.
In Table 11.1 are given typical average values of the coefficient of linear expansion for some materials in the temperature range `0 °C` to `100 °C.` From this Table, compare the value of `α_1` for glass and copper.
We find that copper expands about five times more than glass for the same rise in temperature. Normally, metals expand more and have relatively high values of `α_1.`
Similarly, we consider the fractional change in volume,` (DeltaV)/V` . of a substance for temperature change `ΔT` and define the
coefficient of volume expansion, `alpha_V`
`color{blue}(alpha_V= (DeltaV/V)1/(DeltaT))`......(11.5)
Here `α_V` is also a characteristic of the substance but is not strictly a constant. It depends in general on temperature (Fig 11.6). It is seen that `α_V` becomes constant only at a high temperature.
Table 11.2 gives the values of co-efficient of volume expansion of some common substances in the temperature range `0 –100 °C.` You can see that thermal expansion of these substances (solids and liquids) is rather small, with materials like pyrex glass and invar (a special iron-nickel alloy) having particularly low values of `α_V`.
From this Table we find that the value of `α_v` for alcohol (ethyl) is more than mercury and expands more than mercury for the same rise in temperature.
Water exhibits an anomalous behavour; it contracts on heating between `0 °C` and `4 °C.` The volume of a given amount of water decreases as it is cooled from room temperature, until its temperature reaches `4 °C,` [Fig. 11.7(a)]. Below `4 °C,` the volume increases, and therefore the density decreases [Fig. 11.7(b)].
This means that water has a maximum density at `4 °C.` This property has an important environmental effect: Bodies of water, such as much as order(s) of magnitude larger than the coefficient of volume expansion of typical liquids.
There is a simple relation between the coefficient of volume expansion `(α_v)` and coefficient of linear expansion `(α_l).` Imagine a cube of length, l, that expands equally in all directions, when its temperature increases by `ΔT.` We have
`color{orange}(Δl = α_1 ΔT)`
so, `color{red}(ΔV = (l+Δl)^3 – l^3 = 3l^2 Δl)`..(11.7)
In equation (11.7), terms in `(Δl)^2` and `(Δl)^3` have been neglected since Δl is small compared to l. So
`color{blue}(DeltaV= (3VDeltal)/l = 3Valpha_1DeltaT)`...(11.8)
which gives `color{blue}(alpha_V=3alpha_1)`..........(11.9)
What happens by preventing the thermal expansion of a rod by fixing its ends rigidly? Clearly, the rod acquires a compressive strain due to the external forces provided by the rigid support at the ends.
The corresponding stress set up in the rod is called thermal stress. For example, consider a steel rail of length 5 m and area of cross section `40 cm^2` that is prevented from expanding while the temperature rises by `10 °C.`
The coefficient of linear expansion of steel
is `color{orange}(α_(l("steel")) = 1.2 xx10^(–5) K^(–1))`.
Thus, the compressive strain is `color{orange}((Deltal)/(l) = alpha_(l("steel")) DeltaT= 1.2 xx10^(-5) 1= 1.2\ \ 10^(-4))`
Youngs modulus of steel is `Y_(steel) = 2xx10^11 Nm^(-2)` Therefore, the thermal stress developed is
`color{orange}(F/a = Y _(steel) ((Deltal)/l) = 2.4xx 10^7 \ \N m^(-2))` , which corresponds to an external force of
`color{navy}(DeltaF = AY _(steel) ((Deltal)/l) = 2.4 xx 10^7 \ \40 \ \ 10^(-4) = 10^5N)`
If two such steel rails, fixed at their outer ends, are in contact at their inner ends, a force of this magnitude can easily bend the rails.
You may have observed that sometimes sealed bottles with metallic lids are so tightly screwed that one has to put the lid in hot water for sometime to open the lid. This would allow the metallic cover to expand, thereby loosening it to unscrew easily.
In case of liquids, you may have observed that mercury in a thermometer rises, when the thermometer is put in a slightly warm water. If we take out the thermometer from the warm water the level of mercury falls again.
Similarly, in the case of gases, a balloon partially inflated in a cool room may expand to full size when placed in warm water. On the other hand, a fully inflated balloon when immersed in cold water would start shrinking due to contraction of the air inside.
It is our common experience that most substances expand on heating and contract on cooling. A change in the temperature of a body causes change in its dimensions. The increase in the dimensions of a body due to the increase in its temperature is called `"thermal expansion."`
The expansion in length is called linear expansion. The expansion in area is called area expansion. The expansion in volume is called volume expansion (Fig. 11.5).
If the substance is in the form of a long rod, then for small change in temperature, `ΔT,` the fractional change in length, `Δl//l,` is directly proportional to `ΔT.`
`color{orange}((Deltal)/l = alpha_1DeltaT)`.......(11.4)
where `α_1` is known as the coefficient of linear expansion and is characteristic of the material of the rod.
In Table 11.1 are given typical average values of the coefficient of linear expansion for some materials in the temperature range `0 °C` to `100 °C.` From this Table, compare the value of `α_1` for glass and copper.
We find that copper expands about five times more than glass for the same rise in temperature. Normally, metals expand more and have relatively high values of `α_1.`
Similarly, we consider the fractional change in volume,` (DeltaV)/V` . of a substance for temperature change `ΔT` and define the
coefficient of volume expansion, `alpha_V`
`color{blue}(alpha_V= (DeltaV/V)1/(DeltaT))`......(11.5)
Here `α_V` is also a characteristic of the substance but is not strictly a constant. It depends in general on temperature (Fig 11.6). It is seen that `α_V` becomes constant only at a high temperature.
Table 11.2 gives the values of co-efficient of volume expansion of some common substances in the temperature range `0 –100 °C.` You can see that thermal expansion of these substances (solids and liquids) is rather small, with materials like pyrex glass and invar (a special iron-nickel alloy) having particularly low values of `α_V`.
From this Table we find that the value of `α_v` for alcohol (ethyl) is more than mercury and expands more than mercury for the same rise in temperature.
Water exhibits an anomalous behavour; it contracts on heating between `0 °C` and `4 °C.` The volume of a given amount of water decreases as it is cooled from room temperature, until its temperature reaches `4 °C,` [Fig. 11.7(a)]. Below `4 °C,` the volume increases, and therefore the density decreases [Fig. 11.7(b)].
This means that water has a maximum density at `4 °C.` This property has an important environmental effect: Bodies of water, such as much as order(s) of magnitude larger than the coefficient of volume expansion of typical liquids.
There is a simple relation between the coefficient of volume expansion `(α_v)` and coefficient of linear expansion `(α_l).` Imagine a cube of length, l, that expands equally in all directions, when its temperature increases by `ΔT.` We have
`color{orange}(Δl = α_1 ΔT)`
so, `color{red}(ΔV = (l+Δl)^3 – l^3 = 3l^2 Δl)`..(11.7)
In equation (11.7), terms in `(Δl)^2` and `(Δl)^3` have been neglected since Δl is small compared to l. So
`color{blue}(DeltaV= (3VDeltal)/l = 3Valpha_1DeltaT)`...(11.8)
which gives `color{blue}(alpha_V=3alpha_1)`..........(11.9)
What happens by preventing the thermal expansion of a rod by fixing its ends rigidly? Clearly, the rod acquires a compressive strain due to the external forces provided by the rigid support at the ends.
The corresponding stress set up in the rod is called thermal stress. For example, consider a steel rail of length 5 m and area of cross section `40 cm^2` that is prevented from expanding while the temperature rises by `10 °C.`
The coefficient of linear expansion of steel
is `color{orange}(α_(l("steel")) = 1.2 xx10^(–5) K^(–1))`.
Thus, the compressive strain is `color{orange}((Deltal)/(l) = alpha_(l("steel")) DeltaT= 1.2 xx10^(-5) 1= 1.2\ \ 10^(-4))`
Youngs modulus of steel is `Y_(steel) = 2xx10^11 Nm^(-2)` Therefore, the thermal stress developed is
`color{orange}(F/a = Y _(steel) ((Deltal)/l) = 2.4xx 10^7 \ \N m^(-2))` , which corresponds to an external force of
`color{navy}(DeltaF = AY _(steel) ((Deltal)/l) = 2.4 xx 10^7 \ \40 \ \ 10^(-4) = 10^5N)`
If two such steel rails, fixed at their outer ends, are in contact at their inner ends, a force of this magnitude can easily bend the rails.