Calorimetry
Measuring heat: specific heats, latent heats, and the exchange rate between calories and joules.
Heat, Units & the Calorimeter
What we're measuring, in which units, and the one rule that does the measuring.
Calorimetry is the bookkeeping of heat: measuring how much energy flows when hot and cold things meet. First, what heat is, and the units we count it in.
Heat is energy in transit
When bodies at different temperatures touch, energy flows from the hotter to the colder. Energy transferred because of a temperature difference is heat. Once it arrives, it is part of the receiving body’s internal energy.
Units: the joule and the calorie
Being energy, heat is measured in joules. An older unit, devised before heat was known to be energy, is still widely used:
A 1200 kg car moves at 54 km/h. Express its kinetic energy in kilocalories.
The principle of calorimetry
A simple calorimeter is a copper vessel with a copper stirrer, kept in a wooden box to insulate it, plus a thermometer. Objects at different temperatures are brought together inside it. If no heat is exchanged with the surroundings:
- 1Heat is energy transferred because of a temperature difference; afterwards it is internal energy.
- 21 cal = 4.186 J; 1 kcal = 1000 cal.
- 3Calorimetry: in an insulated calorimeter, heat lost by hot bodies = heat gained by cold ones.
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Specific Heat Capacity
Why iron heats up nine times faster than water, and how to measure it.
Heat a kilogram of water and a kilogram of iron with the same burner and the iron gets hot nine times faster. Each material has its own appetite for heat.
Specific heat capacity
The heat needed to change a body’s temperature depends on its mass, the temperature change, its material and conditions such as pressure:
| Material | cal g⁻¹ °C⁻¹ | J kg⁻¹ K⁻¹ |
|---|---|---|
| Water | 1.00 | 4186 |
| Ice | 0.50 | 2093 |
| Steam | 0.46 | 1926 |
| Aluminium | 0.215 | 900 |
| Iron | 0.112 | 470 |
| Copper | 0.093 | 389 |
| Mercury | 0.033 | 138 |
| Lead | 0.031 | 130 |
(Values at constant pressure, as for solids and liquids heated in the open air.)
How much heat raises an 80 g aluminium block by 25°C? (s = 0.215 cal g⁻¹ °C⁻¹.)
Molar heat capacity, heat capacity and water equivalent
- Counting the amount in moles instead: Q = nCΔθ, where C is the molar heat capacity.
- The product ms is the body’s heat capacity (J/K).
Measuring specific heat: Regnault’s method
For a solid denser than water and insoluble in it. The solid hangs in a double-walled steam chamber until its temperature (θ₁) is steady. A calorimeter of water at θ₂ is slid under it, the solid dropped in, and the highest temperature θ reached is noted. Heat lost by the solid = heat gained by calorimeter and water:
The same apparatus finds a liquid’s specific heat: use a solid of known s₁ and the liquid in place of water, then solve for s₃.
copper calorimeter, 50 g
A 100 g metal ball at 100°C is dropped into 150 g of water at 20.0°C in a 50 g copper calorimeter. The mixture settles at 25.4°C. Find the metal’s specific heat (in cal g⁻¹ °C⁻¹).
- 1Q = msΔθ; s is specific heat capacity. Q = nCΔθ with molar heat capacity C.
- 2Heat capacity = ms; water equivalent = mass of water with the same heat capacity.
- 3Regnault's method: m₁s₁(θ₁ − θ) = (m₂s₂ + m₃s₃)(θ − θ₂).
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Latent Heat
Heat that changes the phase instead of the temperature.
A pot of water on a stove sits at 100°C for many minutes while it boils away, though the flame pours in heat the whole time. Where does that heat go?
Latent heat
Heat can change a body’s phase instead of its temperature. While a solid melts or a liquid boils, the temperature stays fixed. The heat needed is
Why the temperature stays put
In a solid, strong forces hold molecules nearly in place. In a liquid they are weaker: molecules roam inside but can’t leave the surface. In a vapour the forces are almost nil. Melting and boiling pull molecules apart against attraction, which takes energy without speeding them up. So a body’s internal energy is larger as a liquid than as a solid, and larger again as a vapour.
How much heat turns 0.50 kg of ice at −20°C into steam at 100°C? (s_ice 2100, s_water 4200 J kg⁻¹ K⁻¹; L_f 3.36 × 10⁵, L_v 2.25 × 10⁶ J/kg.)
80 g of ice at 0°C is dropped into 300 g of water at 20°C. Take L = 80 cal/g. What happens?
Measuring the latent heat of fusion of ice
Weigh a calorimeter and stirrer (m₁), add water (m₂) at θ₁. Drop in dried pieces of melting ice, stirring, until the temperature falls to its lowest, θ₂ (all ice melted). Weigh again for the ice mass m₃. Heat lost by calorimeter and water = heat to melt the ice and warm it from 0°C to θ₂:
Measuring the latent heat of vaporization of water
Steam from a boiler at θ₁ passes through a steam trap (to remove condensed drops) and bubbles into water at θ₂ in a calorimeter until it warms by about 5°C to θ₃. Weighing gives the mass m₃ of steam condensed. Heat gained by calorimeter and water = heat released by condensing steam plus its cooling from θ₁ to θ₃:
A calorimeter of water equivalent 20 g holds 180 g of water at 20°C. Steam at 100°C is bubbled in until the temperature is 26°C; the mass grows by 2.0 g. Find L (cal/g).
- 1Q = mL for a change of phase at constant temperature; released on freezing or condensing.
- 2Melting and boiling separate molecules against attraction: internal energy rises without a temperature rise.
- 3If the available heat can't melt all the ice, the mixture stays at 0°C.
- 4Fusion: L = (m₁s₁ + m₂s₂)(θ₁ − θ₂)/m₃ − s₂θ₂. Vaporization: (m₁s₁ + m₂s₂)(θ₃ − θ₂) = m₃L + m₃s₂(θ₁ − θ₃).
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The Mechanical Equivalent of Heat
Warming water by friction alone shows a calorie is 4.186 joules.
For a long time heat and work were measured in unrelated units, calories and joules. Experiments that warmed water purely by doing work proved they are the same thing and fixed the exchange rate.
The mechanical equivalent of heat
If mechanical work W produces the same temperature change as heat H:
Searle’s cone method
Two cones of the same metal fit snugly one inside the other. The outer cone A is spun on a spindle (by a motor or by hand), and a counter records its revolutions. The inner cone B, holding water, a thermometer and a stirrer, has a wooden disc of radius r on top with a cord wound round it, running over a pulley to a pan of mass M.
Friction tries to drag B round with A. Choose the direction so the hanging weight’s torque Mgr opposes the friction torque Γ, and adjust M until B stays still: Γ = Mgr. In n revolutions, the work done against friction is
All of it raises the temperature of the water (m₁, s₁) and the cones (m₂, s₂) from θ₁ to θ₂:
With r = 5.0 cm and M = 2.0 kg, the outer cone turns 500 times. The inner cone holds 200 g of water; the cones have a water equivalent of 14 g. The water warms by 3.44°C. Find J.
- 1W = JH; J = 4.186 J/cal.
- 2Searle's cone: friction work 2πnMgr, balanced so the inner cone is still, warms water and cones.
- 3J = 2πnMgr/[(m₁s₁ + m₂s₂)(θ₂ − θ₁)].