Lecture 48 Phase Transition Phases: gas, liquid, solid Phases transition Phase diagram Clausius-Clapeyron equation

State of matter Classically, states of matter are distinguished by changes in specific heat capacity, pressure and temperature. States are distinguished by a discontinuity in one of those properties Example: raising the temperature of ice produces a clear discontinuity at 0℃ as energy goes into phase transition, instead of temperature increase. Four states of matter are observable in everyday life: solid, liquid, gas, and plasma.

Solid In a solid the particles (ions, atoms or molecules) are closely packed together. The forces between particles are strong so that the particles cannot move freely but can only vibrate. A solid has a stable, definite shape, and a definite volume. Solids can only change their shape by force, as when broken or cut.

Liquid A liquid is a nearly incompressible fluid that conforms to the shape of its container but retains a (nearly) constant volume independent of pressure. The volume is definite if the temperature and pressure are constant. Intermolecular (or interatomic or interionic) forces are still important, but the molecules have enough energy to move relative to each other and the structure is mobile. This means that the shape of a liquid is not definite but is determined by its container.

Gas A gas is a compressible fluid. Not only will a gas conform to the shape of its container but it will also expand to fill the container. In a gas, the molecules have enough kinetic energy so that the effect of intermolecular forces is small (or zero for an ideal gas), and the typical distance between neighboring molecules is much greater than the molecular size. A gas has no definite shape or volume, but occupies the entire container in which it is confined.

Plasma Like a gas, plasma does not have definite shape or volume. Unlike gases, plasmas are electrically conductive, produce magnetic fields and electric currents, and respond strongly to electromagnetic forces. Positively charged nuclei swim in a "sea" of freely- moving disassociated electrons, similar to the way such charges exist in conductive metal.

Phase transition A phase transition is the transformation of a thermodynamic system from one phase or state of matter to another one by heat transfer.

Phase diagram A phase diagram is a type of chart used to show conditions at which thermodynamically distinct phases can occur at equilibrium. A typical phase diagram. The dotted green line gives the anomalous behavior of water.

Pressure–temperature phase diagram of water

Latent heat Latent heat is the energy released or absorbed by a body or a thermodynamic system during a constant-temperature process. A typical example is a change of state of matter, meaning a phase transition. The latent heat of condensation of water in the temperature range from −25℃ to 40 ℃ is approximated by the following empirical cubic function: For sublimation and deposition from and into ice, the latent heat is almost constant in the temperature range from −40℃ to 0 ℃ and can be approximated by the following empirical quadratic function: where the temperature  is taken to be the numerical value in ℃

Design a Carnot’s cycle Isothermal process at high temperature 𝑇. By increasing volume, liquid will evaporate to vapor, but the pressure remains constant. Keep increasing the volume till all the liquid evaporate. Decrease the pressure to 𝑝−Δ𝑝 via adiabatic process, the temperature changes to 𝑇−Δ𝑇 Decrease the volume and keep the temperature constant till all the vapor changes to liquid Increase the pressure let the system go back to the initial state. The cycle is different from Carnot’s cycle slightly. But we can neglect it Liquid Vapor L-V 𝑉 𝐿 𝑉 𝐺

Let 𝐿 be the heat needed to vaporize the substance The work done is Δ𝑊=Δ𝑝Δ𝑉 So 𝜂= Δ𝑊 𝐿 = Δ𝑝Δ𝑉 𝐿 = ΔT T Or Δ𝑝 Δ𝑇 = 𝐿 𝑇Δ𝑉 = 𝐿 𝑇 𝑉 𝐺 − 𝑉 𝐿 Take Δ𝑇→0, we have 𝑑𝑝 𝑑𝑇 = 𝐿 𝑇 𝑉 𝐺 − 𝑉 𝐿

Clausius-Clapeyron equation It is a way of characterizing a discontinuous phase transition between two phases of matter of a single constituent. 𝑑𝑝 𝑑𝑇 = 𝐿 𝑇 𝑉 2 − 𝑉 1

For Liquid-Gas phase transition 𝑉 𝐺 ≫ 𝑉 𝐿 , 𝑉 𝐺 − 𝑉 𝐿 ≈ 𝑉 𝐺 =𝑛𝑅𝑇/𝑝 the equation will be 𝑑𝑝 𝑑𝑇 = 𝐿 𝑇 𝑉 𝐺 = 𝐿𝑝 𝑛𝑅 𝑇 2 Or 𝑑𝑝 𝑝 = 𝐿 𝑛𝑅 𝑇 2 𝑑𝑇=− 𝐿 𝑛𝑅 𝑑 1 𝑇 The solution is 𝑝= 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 𝑒 − 𝐿 𝑛𝑅𝑇