Measuring Cp and Cv
A long stream of gas for Cp; a delicate steam balance for Cv.
Lesson at a glanceShow the key points
- Regnault: gas at constant pressure flows through a hot coil (θ₁) then a calorimeter coil; nCp[θ₁ − (θ₂ + θ₃)/2] = (W + m)s(θ₃ − θ₂).
- n from the tank's pressure drop: n = (p₁ − p₂)V/RT.
- Joly: two equal copper spheres in a steam chamber on a balance; the extra steam m₂ condensing on the gas-filled one: Cv = Mm₂L/[m₁(θ₂ − θ₁)].
Gases are light, so they hold little heat. Measuring their heat capacities takes patience: either a long stream of gas, or a very sensitive balance.
Cₚ by Regnault’s method
Gas stored at high pressure in a large tank (kept at constant temperature in water) flows through a copper coil in a hot oil bath at θ₁, then through a second coil in a calorimeter of water. A screw valve is adjusted continuously so that a manometer after it shows constant pressure: the gas flows at constant pressure. The water warms from θ₂ to θ₃.
The gas enters the calorimeter coil at θ₁ and leaves at a temperature that rises from θ₂ to θ₃, on average (θ₂ + θ₃)/2. With W the water equivalent of calorimeter and coil, m the water and s its specific heat:
How many moles passed?
A manometer on the tank gives its pressure before (p₁) and after (p₂). For the tank of volume V at temperature T:
Cᵥ by Joly’s differential steam calorimeter
Two identical hollow copper spheres hang from the arms of a sensitive balance inside a steam chamber, each with a small pan below to catch condensed water. Both are evacuated and balanced; then gas (mass m₁) is put into one and the balance restored with weights.
Steam is let in. It condenses on both spheres, heating them from θ₁ to θ₂, but more condenses on the sphere with gas, because it must also warm the gas. The extra condensed mass m₂ is found by rebalancing. Heated wire coils stop drops forming on the suspensions, and small umbrella shields keep drips off. Since the volume is constant:
(M = molar mass; the spheres’ small expansion is neglected.)
0.80 g of air (29 g/mol) is warmed from 20°C to 100°C in a Joly calorimeter. The extra steam condensed is 22 mg; L = 2.26 × 10⁶ J/kg. Find Cᵥ.
- 1Regnault (Cₚ): gas at constant pressure carries heat from a hot bath into a calorimeter; nCₚ[θ₁ − (θ₂ + θ₃)/2] = (W + m)s(θ₃ − θ₂); n from the tank's pressure drop.
- 2Joly (Cᵥ): the extra steam condensing on a gas-filled sphere: Cᵥ = Mm₂L/[m₁(θ₂ − θ₁)].
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