# 6.2 Thermal expansion of solids and liquids  (Page 5/9)

 Page 5 / 9

## Section summary

• Thermal expansion is the increase, or decrease, of the size (length, area, or volume) of a body due to a change in temperature.
• Thermal expansion is large for gases, and relatively small, but not negligible, for liquids and solids.
• Linear thermal expansion is
$\text{Δ}L=\mathrm{\alpha L}\text{Δ}T,$
where $\text{Δ}L$ is the change in length $L$ , $\text{Δ}T$ is the change in temperature, and $\alpha$ is the coefficient of linear expansion, which varies slightly with temperature.
• The change in area due to thermal expansion is
$\text{Δ}A=2\mathrm{\alpha A}\text{Δ}T,$
where $\text{Δ}A$ is the change in area.
• The change in volume due to thermal expansion is
$\text{Δ}V=\mathrm{\beta V}\text{Δ}T,$
where $\beta$ is the coefficient of volume expansion and $\beta \approx 3\alpha$ . Thermal stress is created when thermal expansion is constrained.

## Conceptual questions

Thermal stresses caused by uneven cooling can easily break glass cookware. Explain why Pyrex®, a glass with a small coefficient of linear expansion, is less susceptible.

Water expands significantly when it freezes: a volume increase of about 9% occurs. As a result of this expansion and because of the formation and growth of crystals as water freezes, anywhere from 10% to 30% of biological cells are burst when animal or plant material is frozen. Discuss the implications of this cell damage for the prospect of preserving human bodies by freezing so that they can be thawed at some future date when it is hoped that all diseases are curable.

One method of getting a tight fit, say of a metal peg in a hole in a metal block, is to manufacture the peg slightly larger than the hole. The peg is then inserted when at a different temperature than the block. Should the block be hotter or colder than the peg during insertion? Explain your answer.

Does it really help to run hot water over a tight metal lid on a glass jar before trying to open it? Explain your answer.

Liquids and solids expand with increasing temperature, because the kinetic energy of a body’s atoms and molecules increases. Explain why some materials shrink with increasing temperature.

## Problems&Exercises

The height of the Washington Monument is measured to be 170 m on a day when the temperature is $\text{35}\text{.}0\text{º}\text{C}$ . What will its height be on a day when the temperature falls to $–\text{10}\text{.}0\text{º}\text{C}$ ? Although the monument is made of limestone, assume that its thermal coefficient of expansion is the same as marble’s.

169.98 m

How much taller does the Eiffel Tower become at the end of a day when the temperature has increased by $\text{15}\text{º}\text{C}$ ? Its original height is 321 m and you can assume it is made of steel.

What is the change in length of a 3.00-cm-long column of mercury if its temperature changes from $\text{37}\text{.}0\text{º}\text{C}$ to $\text{40}\text{.}0\text{º}\text{C}$ , assuming the mercury is unconstrained?

$5\text{.}4×{\text{10}}^{-6}\phantom{\rule{0.25em}{0ex}}\text{m}$

How large an expansion gap should be left between steel railroad rails if they may reach a maximum temperature $\text{35}\text{.}0\text{º}\text{C}$ greater than when they were laid? Their original length is 10.0 m.

You are looking to purchase a small piece of land in Hong Kong. The price is “only” \$60,000 per square meter! The land title says the dimensions are $\text{20}\phantom{\rule{0.25em}{0ex}}\text{m}\phantom{\rule{0.20em}{0ex}}×\phantom{\rule{0.20em}{0ex}}\text{30 m}\text{.}$ By how much would the total price change if you measured the parcel with a steel tape measure on a day when the temperature was $\text{20}\text{º}\text{C}$ above normal?

Because the area gets smaller, the price of the land DECREASES by $\text{~}\text{17},\text{000}\text{.}$

Global warming will produce rising sea levels partly due to melting ice caps but also due to the expansion of water as average ocean temperatures rise. To get some idea of the size of this effect, calculate the change in length of a column of water 1.00 km high for a temperature increase of $1\text{.}\text{00}\text{º}\text{C}\text{.}$ Note that this calculation is only approximate because ocean warming is not uniform with depth.

Show that 60.0 L of gasoline originally at $\text{15}\text{.}0\text{º}\text{C}$ will expand to 61.1 L when it warms to $\text{35}\text{.}0\text{º}\text{C,}$ as claimed in [link] .

$\begin{array}{lll}V& =& {V}_{0}+\text{Δ}V={V}_{0}\left(1+\beta \text{Δ}T\right)\\ & =& \left(\text{60}\text{.}\text{00 L}\right)\left[1+\left(\text{950}×{\text{10}}^{-6}/\text{º}\text{C}\right)\left(\text{35}\text{.}0\text{º}\text{C}-\text{15}\text{.}0\text{º}\text{C}\right)\right]\\ & =& \text{61}\text{.}1\phantom{\rule{0.25em}{0ex}}\text{L}\end{array}$

(a) Suppose a meter stick made of steel and one made of invar (an alloy of iron and nickel) are the same length at $0\text{º}\text{C}$ . What is their difference in length at $\text{22}\text{.}0\text{º}\text{C}$ ? (b) Repeat the calculation for two 30.0-m-long surveyor’s tapes.

(a) If a 500-mL glass beaker is filled to the brim with ethyl alcohol at a temperature of $5\text{.}\text{00}\text{º}\text{C,}$ how much will overflow when its temperature reaches $\text{22}\text{.}0\text{º}\text{C}$ ? (b) How much less water would overflow under the same conditions?

(a) 9.35 mL

(b) 7.56 mL

Most automobiles have a coolant reservoir to catch radiator fluid that may overflow when the engine is hot. A radiator is made of copper and is filled to its 16.0-L capacity when at $\text{10}\text{.}0º\text{C}\text{.}$ What volume of radiator fluid will overflow when the radiator and fluid reach their $\text{95}\text{.}0º\text{C}$ operating temperature, given that the fluid’s volume coefficient of expansion is $\beta =\text{400}×{\text{10}}^{–6}/\text{º}\text{C}$ ? Note that this coefficient is approximate, because most car radiators have operating temperatures of greater than $\text{95}\text{.}0\text{º}\text{C}\text{.}$

A physicist makes a cup of instant coffee and notices that, as the coffee cools, its level drops 3.00 mm in the glass cup. Show that this decrease cannot be due to thermal contraction by calculating the decrease in level if the $\text{350}\phantom{\rule{0.25em}{0ex}}{\text{cm}}^{3}$ of coffee is in a 7.00-cm-diameter cup and decreases in temperature from $\text{95}\text{.}0\text{º}\text{C}\phantom{\rule{0.25em}{0ex}}$ to $\phantom{\rule{0.25em}{0ex}}\text{45}\text{.}0\text{º}\text{C}\text{.}$ (Most of the drop in level is actually due to escaping bubbles of air.)

0.832 mm

(a) The density of water at $0\text{º}\text{C}$ is very nearly $\text{1000}\phantom{\rule{0.25em}{0ex}}{\text{kg/m}}^{3}$ (it is actually $9\text{99}\text{.}{\text{84 kg/m}}^{3}$ ), whereas the density of ice at $0\text{º}\text{C}$ is $9{\text{17 kg/m}}^{3}$ . Calculate the pressure necessary to keep ice from expanding when it freezes, neglecting the effect such a large pressure would have on the freezing temperature. (This problem gives you only an indication of how large the forces associated with freezing water might be.) (b) What are the implications of this result for biological cells that are frozen?

Show that $\beta \approx 3\alpha ,$ by calculating the change in volume $\text{Δ}V$ of a cube with sides of length $L\text{.}$

We know how the length changes with temperature: $\text{Δ}L={\mathrm{\alpha L}}_{0}\text{Δ}T$ . Also we know that the volume of a cube is related to its length by $V={L}^{3}$ , so the final volume is then $V={V}_{0}+\text{Δ}V={\left({L}_{0}+\text{Δ}L\right)}^{3}$ . Substituting for $\text{Δ}L$ gives

$V={\left({L}_{0}+{\mathrm{\alpha L}}_{0}\text{Δ}T\right)}^{3}={L}_{0}^{3}{\left(1+\alpha \text{Δ}T\right)}^{3}\text{.}$

Now, because $\alpha \text{Δ}T$ is small, we can use the binomial expansion:

$V\approx {L}_{0}^{3}\left(1+3\alpha \Delta T\right)={L}_{0}^{3}+3\alpha {L}_{0}^{3}\text{Δ}T.$

So writing the length terms in terms of volumes gives $V={V}_{0}+\text{Δ}V\approx {V}_{0}+{3\alpha V}_{0}\text{Δ}T,$ and so

$\text{Δ}V={\mathrm{\beta V}}_{0}\text{Δ}T\approx {3\alpha V}_{0}\text{Δ}T,\phantom{\rule{0.25em}{0ex}}\text{or}\phantom{\rule{0.25em}{0ex}}\beta \approx 3\alpha .$

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s.
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Tarell
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Damian
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Tarell
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CYNTHIA
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s.
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