# 1.3 Thermal expansion  (Page 3/10)

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## Thermal expansion in three dimensions

The relationship between volume and temperature $\frac{dV}{dT}$ is given by $\frac{dV}{dT}=\beta V\text{Δ}T$ , where $\beta$ is the coefficient of volume expansion    . As you can show in [link] , $\beta =3\alpha$ . This equation is usually written as

$\text{Δ}V=\beta V\text{Δ}T.$

Note that the values of $\beta$ in [link] are equal to $3\alpha$ except for rounding.

Volume expansion is defined for liquids, but linear and area expansion are not, as a liquid’s changes in linear dimensions and area depend on the shape of its container. Thus, [link] shows liquids’ values of $\beta$ but not $\alpha$ .

In general, objects expand with increasing temperature. Water is the most important exception to this rule. Water does expand with increasing temperature (its density decreases ) at temperatures greater than $4\phantom{\rule{0.2em}{0ex}}\text{°}\text{C}\phantom{\rule{0.2em}{0ex}}\left(40\phantom{\rule{0.2em}{0ex}}\text{°}\text{F}\right)$ . However, it is densest at $+4\phantom{\rule{0.2em}{0ex}}\text{°}\text{C}$ and expands with decreasing temperature between $+4\phantom{\rule{0.2em}{0ex}}\text{°}\text{C}$ and $0\phantom{\rule{0.2em}{0ex}}\text{°}\text{C}$ ( $40\phantom{\rule{0.2em}{0ex}}\text{°}\text{F}\phantom{\rule{0.2em}{0ex}}\text{to}\phantom{\rule{0.2em}{0ex}}32\phantom{\rule{0.2em}{0ex}}\text{°}\text{F}$ ), as shown in [link] . A striking effect of this phenomenon is the freezing of water in a pond. When water near the surface cools down to $4\phantom{\rule{0.2em}{0ex}}\text{°}\text{C,}$ it is denser than the remaining water and thus sinks to the bottom. This “turnover” leaves a layer of warmer water near the surface, which is then cooled. However, if the temperature in the surface layer drops below $4\phantom{\rule{0.2em}{0ex}}\text{°}\text{C}$ , that water is less dense than the water below, and thus stays near the top. As a result, the pond surface can freeze over. The layer of ice insulates the liquid water below it from low air temperatures. Fish and other aquatic life can survive in $4\phantom{\rule{0.2em}{0ex}}\text{°}\text{C}$ water beneath ice, due to this unusual characteristic of water.

## Calculating thermal expansion

Suppose your 60.0-L $\left(15.9\phantom{\rule{0.2em}{0ex}}\text{-gal}$ -gal) steel gasoline tank is full of gas that is cool because it has just been pumped from an underground reservoir. Now, both the tank and the gasoline have a temperature of $15.0\phantom{\rule{0.2em}{0ex}}\text{°}\text{C}\text{.}$ How much gasoline has spilled by the time they warm to $35.0\phantom{\rule{0.2em}{0ex}}\text{°}\text{C}$ ?

## Strategy

The tank and gasoline increase in volume, but the gasoline increases more, so the amount spilled is the difference in their volume changes. We can use the equation for volume expansion to calculate the change in volume of the gasoline and of the tank. (The gasoline tank can be treated as solid steel.)

## Solution

1. Use the equation for volume expansion to calculate the increase in volume of the steel tank:
$\text{Δ}{V}_{\text{s}}={\beta }_{\text{s}}{V}_{\text{s}}\text{Δ}T.$
2. The increase in volume of the gasoline is given by this equation:
$\text{Δ}{V}_{\text{gas}}={\beta }_{\text{gas}}{V}_{\text{gas}}\text{Δ}T.$
3. Find the difference in volume to determine the amount spilled as
${V}_{\text{spill}}=\text{Δ}{V}_{\text{gas}}-\text{Δ}{V}_{\text{s}}.$

Alternatively, we can combine these three equations into a single equation. (Note that the original volumes are equal.)

$\begin{array}{cc}\hfill {V}_{\text{spill}}& =\left({\beta }_{\text{ga}}{}_{\text{s}}-{\beta }_{\text{s}}\right)V\text{Δ}T\hfill \\ & =\left[\left(950-35\right)\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-6}\text{/}\text{°}\text{C}\right]\left(60.0\phantom{\rule{0.2em}{0ex}}\text{L}\right)\left(20.0\phantom{\rule{0.2em}{0ex}}\text{°}\text{C}\right)\hfill \\ & =1.10\phantom{\rule{0.2em}{0ex}}\text{L}\text{.}\hfill \end{array}$

## Significance

This amount is significant, particularly for a 60.0-L tank. The effect is so striking because the gasoline and steel expand quickly. The rate of change in thermal properties is discussed later in this chapter.

If you try to cap the tank tightly to prevent overflow, you will find that it leaks anyway, either around the cap or by bursting the tank. Tightly constricting the expanding gas is equivalent to compressing it, and both liquids and solids resist compression with extremely large forces. To avoid rupturing rigid containers, these containers have air gaps, which allow them to expand and contract without stressing them.

#### Questions & Answers

Newton's second laws is call with
Dyutee Reply
Really
Arzoodan
what is mean by thermodynamics
Prasad Reply
it is study about temperature and it's equilibrium
thiru
Its the study of heat and its relation with others kind of energy
Antonio
state caulombs law clearly
constand Reply
show mathematically that an electron has the greater speed than the proton when they attract each other
ezra Reply
show mathematically that an electron has the greater speed than the proton when they attract each other
srikanta
@ezra & srikanta; for electrons: a=ke^2/(mr^2) and for protons: a=kp^2/(mr^2)
Sikandar
what is electrostatics
Hero Reply
the study of charge at rest
Gulzar
@Hero; the study of charges at rest is the electrostatics
Sikandar
okay what is electrostatic?
Abd
charge at rest
Nawal
set of character...
Arzoodan
oky
Abd
Gauss law, electric fields, dipoles,...
Antonio
good
Abd
A proton initially at rest falls through a p.d of 25000V. what speed does it gain?
Minister Reply
@Minister; use equation v= sq root(2×eV/m)
Sikandar
what is the reaction of heat on magnet
ORIZINO Reply
Magnetization decreases with increase in temperature. But in case of diamagnetic substance heat has no role on magnetization.
srikanta
what is a physical significant of electric dipole moment .
PRANAB Reply
A dipole moment it's a mechanical electrical effect used in nature
Antonio
what is the uses of carbon brushes in generator
Malik Reply
to minimize heat
constand
at what temperature is the degree Fahrenheit equal to degree Celsius
Grace Reply
Celsius and Faharaneith are different, never equal
Antonio
find their liners express of n=a+b/T² ( plot graph n against T)
Donsmart Reply
Radio Stations often advertis "instant news,,if that meens you can hear the news the instant the radio announcer speaks it is the claim true? what approximate time interval is required for a message to travel from Cairo to Aswan by radio waves (500km) (Assume the waves Casbe detected at this range )
mahmod Reply
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Pawan Reply
Can someone please predict the trajectory of a point charge in a uniform electric field????
erlinda Reply
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zahid Reply
how do you convert temperature in degree Celsius to Fahrenheit
kwame
Celsius x 9/5 +32
Cyclone

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