10.1 Temperature, the Ideal-Gas Law and Thermal Expansion
Temperature measures how hot a body is; heat is energy that flows because of a temperature difference. Gases lead us to absolute zero and $pV=nRT$, and almost everything expands when heated: $\Delta L=\alpha L\,\Delta T$.
A doctor reads $38.5\ ^\circ\text{C}$ on a thermometer and knows you have a fever; a weather report from New York says $95\ ^\circ\text{F}$ and means the same kind of heatwave as $35\ ^\circ\text{C}$ in Chennai. Both rely on something so familiar that we rarely question it: a column of liquid in a thin tube that grows longer when it gets hotter. The same expansion that makes thermometers work also buckles railway lines, cracks glass dishes and pops the lids off tight jars when you hold them under hot water.
This section sorts out the basic ideas. We separate temperature from heat, build temperature scales from two fixed points, and see how the pressure of a gas at fixed volume points to a lowest possible temperature — absolute zero — and to the kelvin scale used in every formula of this chapter. Along the way we meet the ideal-gas equation $pV=nRT$. Then we measure thermal expansion: how much a rod lengthens, a plate spreads and a liquid swells for each kelvin, why a bimetallic strip curls, and why water — densest at $4\ ^\circ\text{C}$ — behaves so strangely.
Heat, temperature and temperature scales
Touch a metal railing and a wooden bench on a cold morning: the metal feels colder, although both have been outside all night and are at the same temperature. Your skin is a poor thermometer — it senses how fast energy leaves your hand, not how hot the object is. Physics needs a sharper language.
Temperature is a measure of how hot or cold a body is. When two bodies are placed in contact, energy flows from the one at higher temperature to the one at lower temperature until their temperatures become equal; they are then in thermal equilibrium (you will meet this idea formally in Chapter 11).
Heat $Q$ is the energy transferred between a system and its surroundings because of a temperature difference between them. Heat is energy in transit: its SI unit is the joule, $\text{J}$. The older unit, the calorie, is the heat that raises the temperature of $1\ \text{g}$ of water by $1\ ^\circ\text{C}$ (from $14.5$ to $15.5\ ^\circ\text{C}$): $1\ \text{cal}=4.186\ \text{J}$.
A body does not "contain heat", just as a bank account does not contain "deposits": heat describes energy on its way in or out. A spoonful of boiling water is hotter than a bathtub of warm water, yet the bathtub can melt far more ice — temperature and the amount of energy a body can give up are different things.
Thermometers. Any property that changes steadily with temperature can be used to measure it: the length of a mercury or alcohol column, the pressure of a gas held at fixed volume, the electrical resistance of a platinum wire. To turn the property into numbers we choose two reproducible fixed points and divide the interval between them into equal steps. The traditional pair, both at standard atmospheric pressure, are the ice point (pure ice melting) and the steam point (pure water boiling).
On the Celsius scale the ice point is $0\ ^\circ\text{C}$ and the steam point $100\ ^\circ\text{C}$: $100$ equal divisions. On the Fahrenheit scale the ice point is $32\ ^\circ\text{F}$ and the steam point $212\ ^\circ\text{F}$: $180$ equal divisions.
For any two scales that are linear in the same thermometric property, the fraction of the way from the ice point to the steam point is the same:
$$\frac{X-X_\text{ice}}{X_\text{steam}-X_\text{ice}}=\text{same on every scale}.$$For Celsius ($t_C$) and Fahrenheit ($t_F$):
$$\frac{t_F-32}{180}=\frac{t_C}{100},\qquad t_F=\tfrac95\,t_C+32 .$$A temperature difference converts without the offset: $\Delta t_F=\tfrac95\,\Delta t_C$, so a change of $1\ ^\circ\text{C}$ equals a change of $1.8\ ^\circ\text{F}$.
The ideal-gas equation and absolute temperature
A mercury thermometer and an alcohol thermometer, both calibrated at $0$ and $100\ ^\circ\text{C}$, disagree slightly at $50\ ^\circ\text{C}$, because the two liquids do not expand in exactly the same way. Gases are far better behaved. At low density, every gas obeys the same simple laws, found experimentally in the seventeenth and eighteenth centuries.
For a fixed amount of a gas at low density:
- Boyle's law: at constant temperature, $pV=\text{constant}$ — halve the volume and the pressure doubles.
- Charles' law: at constant pressure, the volume increases linearly with the temperature; every gas, extrapolated, would reach zero volume at the same temperature, about $-273\ ^\circ\text{C}$.
Similarly, at constant volume the pressure increases linearly with temperature, and every gas extrapolates to zero pressure at the same temperature.
The constant-volume gas thermometer. Trap a little gas in a bulb, keep its volume fixed with a mercury column, and record its pressure at several temperatures. The points lie on a straight line. Do it again with a different gas, or with less gas in the bulb: you get different straight lines, but at low pressures all of them extrapolate to $p=0$ at the same temperature, $-273.15\ ^\circ\text{C}$. Nothing can be colder than the temperature at which a gas would exert no pressure at all. This natural zero is the starting point of the absolute scale.
Absolute zero is the lowest possible temperature, $-273.15\ ^\circ\text{C}$. The absolute (kelvin) temperature $T$ is measured from it in steps the same size as the Celsius degree:
$$T=t_C+273.15 ,$$with the SI unit the kelvin, $\text{K}$ (no degree sign). The scale is fixed by assigning the triple point of water — the single temperature and pressure at which ice, liquid water and water vapour coexist — the value $273.16\ \text{K}$ exactly ($0.01\ ^\circ\text{C}$). A constant-volume gas thermometer then measures
$$T=273.16\ \text{K}\times\frac{p}{p_\text{tp}},$$where $p_\text{tp}$ is the gas pressure at the triple point (in the limit of low gas density).
Absolute temperature is never negative, and a temperature difference is the same number in kelvin and in Celsius degrees. Inside any formula that contains $T$ itself — a ratio of temperatures, $pV=nRT$, $T^4$ in Section 10.3 — always use kelvin.
Combining Boyle's and Charles' laws, $\dfrac{pV}{T}$ is constant for a fixed amount of gas, and experiment shows that the constant is proportional to the number of moles. This gives one equation for every gas at low density.
For $n$ moles of a gas at low density, with pressure $p$, volume $V$ and absolute temperature $T$,
$$pV=nRT ,$$where $R=8.31\ \text{J mol}^{-1}\text{K}^{-1}$ is the universal gas constant and $n=\dfrac{m}{M}$ (mass divided by molar mass). A gas that obeys this equation exactly is an ideal gas; real gases come close at low pressure and high temperature, far from liquefying. For a fixed amount of gas, $\dfrac{p_1V_1}{T_1}=\dfrac{p_2V_2}{T_2}$.
Chapter 12 explains why gases obey this law, from the motion of their molecules. For now it is a well-tested experimental fact — and a very useful one.
Thermal expansion of solids
In a solid the atoms vibrate about their positions in a lattice. The force between neighbours is not symmetric: it is much harder to push two atoms together than to pull them apart. As the temperature rises the vibrations grow, the atoms spend more time on the "far" side, and the average spacing increases. Multiplied over billions of atoms in a row, the tiny increase in spacing becomes a measurable growth in length — proportional, for moderate changes, to the original length and to the rise in temperature.
When the temperature of a body changes by a small amount $\Delta T$, its length $L$, area $A$ and volume $V$ change by
$$\Delta L=\alpha L\,\Delta T,$$ $$\Delta A=\beta A\,\Delta T,$$ $$\Delta V=\gamma V\,\Delta T,$$which define the coefficients of linear expansion $\alpha=\dfrac{\Delta L}{L\,\Delta T}$, area expansion $\beta$ and volume expansion $\gamma$. Each has unit $\text{K}^{-1}$ (the same as $(^\circ\text{C})^{-1}$, since only a temperature difference appears). They depend a little on temperature; the values below are for room temperature.
| Solid | $\alpha$ ($10^{-5}\ \text{K}^{-1}$) | Solid | $\alpha$ ($10^{-5}\ \text{K}^{-1}$) |
|---|---|---|---|
| aluminium | 2.3 | steel | 1.2 |
| brass | 1.9 | soda glass | 0.90 |
| copper | 1.7 | invar (Fe–Ni alloy) | 0.12 |
Liquids have no fixed shape, so only $\gamma$ matters for them, and it is larger: about $2.1\times10^{-4}\ \text{K}^{-1}$ for water at $20\ ^\circ\text{C}$, $1.8\times10^{-4}\ \text{K}^{-1}$ for mercury and $1.1\times10^{-3}\ \text{K}^{-1}$ for ethanol.
For a solid that expands equally in all directions (isotropic), and for $\alpha\,\Delta T\ll1$,
$$\beta=2\alpha,\qquad \gamma=3\alpha .$$Every length of the body — including the diameter of a hole cut in it — scales by the same factor $1+\alpha\,\Delta T$.
Why. A cube of side $L$ heated by $\Delta T$ has side $L(1+\alpha\Delta T)$ and volume $L^3(1+\alpha\Delta T)^3$, which multiplies out to $L^3\big(1+3\alpha\Delta T$ $+\,3\alpha^2\Delta T^2+\alpha^3\Delta T^3\big)$. With $\alpha\Delta T\sim10^{-3}$ the squared and cubed terms are a thousand times smaller and are dropped, leaving $\Delta V=3\alpha V\Delta T$. The same argument for a square gives $\beta=2\alpha$. A hole expands exactly as the material that would fill it — a heated washer gets a bigger hole, which is why a tight metal lid loosens under hot water.
Thermal stress. If a rod is prevented from expanding — welded rails, a pipe fixed between two walls, a concrete slab without joints — the walls must squeeze it back by exactly the length it would have gained. The compressive strain is $\dfrac{\Delta L}{L}=\alpha\Delta T$, and by Hooke's law (Section 8.2) the stress is Young's modulus times that strain.
A rod of Young's modulus $Y$, cross-section $A$ and expansion coefficient $\alpha$, held between rigid supports and heated by $\Delta T$ (within the proportional limit), carries a compressive thermal stress and pushes on the supports with force
$$\sigma=Y\alpha\,\Delta T,\qquad F=YA\alpha\,\Delta T .$$Both are independent of the rod's length. Cooling a rod fixed at both ends gives a tensile stress of the same size.
The bimetallic strip. Rivet a strip of brass to a strip of steel of the same length. On heating, brass ($\alpha=1.9\times10^{-5}\ \text{K}^{-1}$) tries to grow more than steel ($1.2\times10^{-5}\ \text{K}^{-1}$), but the rivets keep them together, so the pair curls into an arc with brass on the outside (the longer, convex side). Cooled below the riveting temperature it curls the other way. A bimetallic strip that bends to open or close an electrical contact is the thermostat in irons, kettles, room heaters and the flashing indicator of older cars; a coiled one turns the pointer of a dial thermometer.
Liquids, gases and the anomalous expansion of water
When a body of fixed mass $m$ expands, its density $\rho=m/V$ falls. Since $V$ becomes $V(1+\gamma\Delta T)$,
$$\rho=\frac{\rho_0}{1+\gamma\,\Delta T}\approx\rho_0\left(1-\gamma\,\Delta T\right),$$the approximation holding when $\gamma\Delta T\ll1$. Warm water rising above cold, hot air rising in a chimney and the convection currents of Section 10.3 all start here. For a gas the expansion is much larger, and the ideal-gas equation gives it at once.
At constant pressure $V=\dfrac{nR}{p}T$, so $\Delta V=\dfrac{nR}{p}\Delta T=\dfrac{V}{T}\Delta T$ and the coefficient of volume expansion is
$$\gamma=\frac1T .$$It is not a constant: $3.66\times10^{-3}\ \text{K}^{-1}$ at $0\ ^\circ\text{C}$, $3.33\times10^{-3}\ \text{K}^{-1}$ at $27\ ^\circ\text{C}$ — some ten times that of a liquid and a hundred times that of a solid.
The anomalous expansion of water. Water breaks the rule. Heated from $0\ ^\circ\text{C}$, it contracts until about $4\ ^\circ\text{C}$ and only then begins to expand; its density is greatest, about $1000\ \text{kg/m}^3$, at $4\ ^\circ\text{C}$ (more precisely $3.98\ ^\circ\text{C}$). Ice is less dense still, about $917\ \text{kg/m}^3$, so water expands by about $9\%$ when it freezes. The reason is the open, hydrogen-bonded structure of ice: some of that structure survives in cold liquid water, and melting it lets the molecules pack more closely — an effect that wins over normal thermal expansion just above $0\ ^\circ\text{C}$.
The consequences are large. As a lake cools in winter, the surface water cools, becomes denser and sinks, until the whole lake is at $4\ ^\circ\text{C}$. Further cooling makes the surface water lighter, so it stays on top, reaches $0\ ^\circ\text{C}$ and freezes. The ice floats and insulates the water below, which stays near $4\ ^\circ\text{C}$ at the bottom: lakes freeze from the top down, and fish survive the winter. The same expansion on freezing bursts water pipes and splits rocks in cold mountains.
1. Celsius inside a ratio. $\dfrac{p_2}{p_1}=\dfrac{T_2}{T_1}$ and $pV=nRT$ need kelvin. Heating a gas from $10\ ^\circ\text{C}$ to $20\ ^\circ\text{C}$ does not double its pressure — it raises it by $\dfrac{10}{283}\approx3.5\%$.
2. Offsets on differences. A rise of $10\ ^\circ\text{C}$ is a rise of $10\ \text{K}$ and $18\ ^\circ\text{F}$ — never add $273$ or $32$ to a difference.
3. Holes shrinking. On heating, a hole in a plate grows, exactly as if it were filled with the same material.
4. $\beta=\alpha^2$ or $\gamma=\alpha^3$. The coefficients add, they do not multiply: $\beta=2\alpha$, $\gamma=3\alpha$ for an isotropic solid.
5. Length in thermal stress. $\sigma=Y\alpha\Delta T$ does not depend on the rod's length or thickness; the force $YA\alpha\Delta T$ depends on the area only.
6. "Water always expands when heated." Not between $0$ and $4\ ^\circ\text{C}$.
In JEE Main, NEET and JEE Advanced (as part of thermal expansion of liquids); not required by the CBSE syllabus.
A liquid is always held in a vessel, and the vessel expands too. What you see — the rise in a thermometer column or the overflow from a full flask — is the apparent expansion, with coefficient
$$\gamma_\text{app}=\gamma_\text{liquid}-\gamma_\text{vessel},$$where $\gamma_\text{vessel}=3\alpha_\text{vessel}$ for a solid container. The overflow from a vessel filled to the brim with volume $V$ is $V\gamma_\text{app}\,\Delta T$.
Example. A soda-glass flask holds exactly $500\ \text{cm}^3$ of mercury at $20\ ^\circ\text{C}$. Heated to $100\ ^\circ\text{C}$: $\gamma_\text{app}=1.8\times10^{-4}$ $-\,3\times0.90\times10^{-5}$ $=1.53\times10^{-4}\ \text{K}^{-1}$, and the overflow is $500\times1.53\times10^{-4}\times80$ $=6.1\ \text{cm}^3$. Ignoring the flask's expansion would give $7.2\ \text{cm}^3$. A mercury thermometer works only because $\gamma_\text{Hg}$ is about seven times $\gamma_\text{glass}$.
Exercises
Use $R=8.31\ \text{J mol}^{-1}\text{K}^{-1}$, $T=t_C+273$ unless more figures are needed, $1\ \text{atm}=1.013\times10^5\ \text{Pa}$, and the expansion coefficients in the table of this section.
- 1 Convert $25\ ^\circ\text{C}$ to degrees Fahrenheit and to kelvin.
- 2 Paper catches fire at about $451\ ^\circ\text{F}$. Express this in degrees Celsius.
- 3 Liquid nitrogen boils at $77\ \text{K}$ at atmospheric pressure. Express this in $^\circ\text{C}$ and $^\circ\text{F}$.
- 4 The temperature of a room rises by $18\ ^\circ\text{F}$. By how many degrees Celsius, and by how many kelvin, has it risen?
- 5 At what temperature is the Fahrenheit reading five times the Celsius reading?
- 6 A thermometer with its own scale $X$ reads $20$ at the ice point and $180$ at the steam point. (a) What is the Celsius temperature when it reads $60$? (b) What does it read at $60\ ^\circ\text{C}$?
- 7 The mercury column of an unmarked thermometer is $4.0\ \text{cm}$ long in melting ice and $24.0\ \text{cm}$ long in steam. Find the temperature when the column is $10.0\ \text{cm}$ long.
- 8 Explain why a temperature can be negative on the Celsius scale but not on the kelvin scale, and why a temperature difference has the same value in both.
- 1 Find the volume occupied by $2.0\ \text{mol}$ of an ideal gas at $27\ ^\circ\text{C}$ and $1.013\times10^5\ \text{Pa}$.
- 2 A rigid oxygen cylinder is filled to $1.50\times10^7\ \text{Pa}$ at $20\ ^\circ\text{C}$. What is its pressure when it is left in a hot van at $50\ ^\circ\text{C}$?
- 3 A balloon holds $3.0\ \text{L}$ of air at $27\ ^\circ\text{C}$. It is dipped in liquid nitrogen at $77\ \text{K}$, the pressure staying constant. Find its new volume (assume the air stays a gas).
- 4 Estimate the mass of air in a classroom $5.0\ \text{m}\times4.0\ \text{m}\times3.0\ \text{m}$ at $27\ ^\circ\text{C}$ and $1.013\times10^5\ \text{Pa}$. The molar mass of air is $0.029\ \text{kg/mol}$.
- 5 A constant-volume gas thermometer reads $5.00\times10^4\ \text{Pa}$ at the triple point of water and $7.30\times10^4\ \text{Pa}$ at the boiling point of a liquid. Find the boiling point in K and $^\circ\text{C}$.
- 6 (Modelling) An air bubble of volume $1.0\ \text{cm}^3$ is released at the bottom of a lake $20\ \text{m}$ deep, where the water is at $7\ ^\circ\text{C}$. Find its volume just below the surface, where the water is at $27\ ^\circ\text{C}$. The pressure at depth $h$ is $p_0+\rho gh$, with $p_0=1.013\times10^5\ \text{Pa}$, $\rho=1.0\times10^3\ \text{kg/m}^3$ and $g=9.8\ \text{m/s}^2$.
- 7 Different gases in a constant-volume gas thermometer give slightly different readings for the same temperature at ordinary pressures. How is a single, gas-independent temperature obtained?
- 1 An aluminium rod is $2.00\ \text{m}$ long at $20\ ^\circ\text{C}$. By how much does it lengthen at $120\ ^\circ\text{C}$?
- 2 A steel bridge is $300\ \text{m}$ long. Its temperature ranges from $-10\ ^\circ\text{C}$ to $45\ ^\circ\text{C}$ over the year. How much room must its expansion joints allow?
- 3 A brass plate has a circular hole of diameter $10.00\ \text{cm}$ at $20\ ^\circ\text{C}$. Find the diameter of the hole at $220\ ^\circ\text{C}$.
- 4 A copper sheet of area $0.50\ \text{m}^2$ is heated through $80\ \text{K}$. Find the increase in its area.
- 5 A steel ball has radius $2.00\ \text{cm}$. Find its volume and the increase in its volume when it is heated through $100\ \text{K}$.
- 6 A steel measuring tape is correct at $25\ ^\circ\text{C}$. On a cold day at $5\ ^\circ\text{C}$ it gives the length of a rod as $85.00\ \text{cm}$. What is the actual length of the rod? (Treat the rod's own expansion as negligible.)
- 7 A steel ring has an inner diameter of $7.980\ \text{cm}$ at $20\ ^\circ\text{C}$. To what temperature must it be heated to slide onto a shaft of diameter $8.000\ \text{cm}$ at $20\ ^\circ\text{C}$?
- 8 A brass rod of cross-section $2.0\ \text{cm}^2$ is fixed between two rigid walls and heated through $40\ \text{K}$. Taking $Y_\text{brass}=1.0\times10^{11}\ \text{Pa}$, find the thermal stress and the force on each wall.
- 1 A car's fuel tank is filled with $50\ \text{L}$ of petrol at $15\ ^\circ\text{C}$ ($\gamma=9.5\times10^{-4}\ \text{K}^{-1}$). The car is parked in the sun and the petrol warms to $35\ ^\circ\text{C}$. Ignoring the expansion of the tank, how much petrol overflows if the tank was full?
- 2 Mercury has density $1.360\times10^4\ \text{kg/m}^3$ at $0\ ^\circ\text{C}$. Find its density at $100\ ^\circ\text{C}$.
- 3 Starting from $pV=nRT$, show that an ideal gas at constant pressure has $\gamma=1/T$, and evaluate it at $27\ ^\circ\text{C}$.
- 4 In a very cold winter, the surface of a deep lake is frozen. What is the temperature (a) just under the ice, (b) at the bottom of the lake? Explain.
- 5 A sealed bottle full of water is warmed from $0\ ^\circ\text{C}$ to $8\ ^\circ\text{C}$. Describe how the volume of the water changes along the way, and compare its volume at $8\ ^\circ\text{C}$ with that at $0\ ^\circ\text{C}$ (use Interactive 2).
- 6 A straight brass–steel bimetallic strip is heated. Which metal ends up on the convex side? What happens if the strip is cooled instead? Why is invar useful in clock parts and measuring tapes?
- 7 (Modelling) Boiling water poured into a cold, thick soda-glass tumbler can crack it, but a borosilicate (Pyrex) dish with $\alpha=3.2\times10^{-6}\ \text{K}^{-1}$ survives. Suppose the inner surface heats by $80\ \text{K}$ while the outer surface stays cold. Estimate the thermal strain and stress in the inner layer for each glass ($Y\approx6.5\times10^{10}\ \text{Pa}$ for both), and compare with the tensile strength of ordinary glass, roughly $5\times10^7\ \text{Pa}$.
E1. Board · 2 marks What is meant by the anomalous expansion of water? Use it to explain why aquatic animals survive in a lake whose surface has frozen.
E2. NEET At what temperature does a Fahrenheit thermometer show a reading twice that of a Celsius thermometer? (a) $80\ ^\circ\text{C}$ (b) $160\ ^\circ\text{C}$ (c) $-24.6\ ^\circ\text{C}$ (d) $320\ ^\circ\text{C}$
E3. JEE Main (Numerical value) A steel rod and a brass rod are to be cut so that the steel rod is always $10.0\ \text{cm}$ longer than the brass rod, at every temperature. With $\alpha_\text{steel}=1.2\times10^{-5}\ \text{K}^{-1}$ and $\alpha_\text{brass}=1.9\times10^{-5}\ \text{K}^{-1}$, find the length of the steel rod in cm, to one decimal place.
E4. JEE Advanced (One or more correct) A uniform rod of length $L$, cross-sectional area $A$, Young's modulus $Y$ and linear expansion coefficient $\alpha$ is fitted without stress between two rigid walls and is then heated through $\Delta T$. Which statements are correct? (A) The stress in the rod is $Y\alpha\Delta T$. (B) The force on each wall is $YA\alpha\Delta T$. (C) A rod of the same material twice as long would carry twice the stress. (D) If one wall is removed so that one end is free, no thermal stress develops.
E5. JEE Main NEET The pressure of an ideal gas in a closed rigid vessel rises by $0.50\%$ when it is heated through $1\ \text{K}$. The initial temperature of the gas was (a) $100\ \text{K}$ (b) $200\ \text{K}$ (c) $273\ \text{K}$ (d) $500\ \text{K}$
E6. Board · Case study In a laboratory, a constant-volume gas thermometer containing helium at low pressure reads $4.00\times10^4\ \text{Pa}$ with its bulb at the triple point of water. (i) The bulb is placed in a bath and the pressure becomes $5.50\times10^4\ \text{Pa}$. Find the temperature of the bath in kelvin and in degrees Celsius. [2] (ii) What pressure would the gas exert at absolute zero if it remained an ideal gas? Why is this point never actually reached? [1] (iii) Why is the gas kept at low pressure? [1]
Research question. How much of the observed rise in sea level comes simply from seawater expanding as the oceans warm?
Why it matters. Rising seas threaten coastal cities from Mumbai and Dhaka to Miami, Lagos and Jakarta, and small island nations in the Pacific and Indian Oceans. Two effects raise the sea: water added by melting glaciers and ice sheets, and thermal expansion of the water already there. Climate scientists separate the two using ocean temperature measurements from thousands of free-drifting profiling floats.
What to do. Model a column of ocean of depth $h$ whose temperature rises by $\Delta T$. Because the sides of the ocean cannot move outward, all of the volume increase shows up as a rise in height.
- Show that a column of height $h$ and fixed base area rises by $\Delta h=\gamma h\,\Delta T$.
- Take $\gamma\approx2.0\times10^{-4}\ \text{K}^{-1}$ for seawater near the surface, and suppose the top $700\ \text{m}$ of the ocean warms by $0.3\ \text{K}$. Estimate $\Delta h$.
- The coefficient for seawater depends strongly on temperature — roughly $0.5\times10^{-4}\ \text{K}^{-1}$ near $0\ ^\circ\text{C}$ and $3\times10^{-4}\ \text{K}^{-1}$ near $25\ ^\circ\text{C}$. Would the same warming raise the sea more in the tropics or near the poles? By roughly what factor?
- The deep ocean is near $2\ ^\circ\text{C}$. Repeat step 2 for a warming of $0.05\ \text{K}$ spread over $3000\ \text{m}$ of deep water, using the cold-water value of $\gamma$.
- Compare your total with published estimates of the "thermosteric" contribution to sea-level rise, and with the total observed rise of the last few decades.
Tools: Calculator or spreadsheet; open ocean-temperature data and sea-level reports from international climate assessments and ocean-observing programmes.
Hand in: A one-page report: your model and its assumptions, the calculations, a comparison table (your estimate versus published values) and two sentences on which assumption matters most.
Going further. Water is not the only thing that expands: find out how engineers allow for thermal expansion in long pipelines, bridges and railways in places with large temperature swings, such as deserts and Himalayan passes.
🧪 Self-check quiz
Q1. The Celsius and Fahrenheit scales give the same reading at
Q2. Heating a metal cube increases each of its edges by $0.10\%$. Its volume increases by about
Q3. A metal plate with a circular hole in it is heated uniformly. The hole
Q4. The density of water is greatest at about
Q5. An ideal gas is heated at constant pressure from $27\ ^\circ\text{C}$ to $327\ ^\circ\text{C}$. Its volume