Thermodynamics

Thermodynamic Systems, States and Processes Objectives are to: define thermodynamics systems and states of systems explain how processes affect such systems apply the above thermodynamic terms and ideas to the laws of thermodynamics

At room temperature, for most gases: Internal Energy of a Classical ideal gas “Classical” means Equipartition Principle applies: each molecule has average energy ½ kT per in thermal equilibrium. At room temperature, for most gases: monatomic gas (He, Ne, Ar, …) 3 translational modes (x, y, z) diatomic molecules (N2, O2, CO, …) 3 translational modes (x, y, z) + 2 rotational modes (wx, wy) Note: At higher temps, other modes can come in, e.g., vibrational modes.

Internal Energy of a Gas A pressurized gas bottle (V = 0.05 m3), contains helium gas (an ideal monatomic gas) at a pressure p = 1×107 Pa and temperature T = 300 K. What is the internal thermal energy of this gas?

Changing the Internal Energy U is a “state” function --- depends uniquely on the state of the system in terms of p, V, T etc. (e.g. For a classical ideal gas, U = NkT ) There are two ways to change the internal energy of a system: WORK done by the system on the environment Wby = -Won HEAT is the transfer of thermal energy into the system from the surroundings Q Thermal reservoir Work and Heat are process energies, not state functions.

Work Done by An Expanding Gas The expands slowly enough to maintain thermodynamic equilibrium. Increase in volume, dV +dV Positive Work (Work is done by the gas) -dV Negative Work (Work is done on the gas)

A Historical Convention +dV Positive Work (Work is done by the gas) -dV Negative Work (Work is done on the gas) Energy leaves the system and goes to the environment. Energy enters the system from the environment.

Total Work Done To evaluate the integral, we must know how the pressure depends (functionally) on the volume.

Pressure as a Function of Volume Work is the area under the curve of a PV-diagram. Work depends on the path taken in “PV space.” The precise path serves to describe the kind of process that took place.

Different Thermodynamic Paths The work done depends on the initial and final states and the path taken between these states.

Work done by a Gas dWby = F dx = pA dx = p (A dx)= p dV dx When a gas expands, it does work on its environment Consider a piston with cross-sectional area A filled with gas. For a small displacement dx, the work done by the gas is: dx dWby = F dx = pA dx = p (A dx)= p dV We generally assume quasi-static processes (slow enough that p and T are well defined at all times): This is just the area under the p-V curve V p p V p V Note that the amount of work needed to take the system from one state to another is not unique! It depends on the path taken.

What is Heat? Up to mid-1800’s heat was considered a substance -- a “caloric fluid” that could be stored in an object and transferred between objects. After 1850, kinetic theory. A more recent and still common misconception is that heat is the quantity of thermal energy in an object. The term Heat (Q) is properly used to describe energy in transit, thermal energy transferred into or out of a system from a thermal reservoir … (like cash transfers into and out of your bank account) Q U Q is not a “state” function --- the heat depends on the process, not just on the initial and final states of the system Sign of Q : Q > 0 system gains thermal energy Q < 0 system loses thermal energy

An Extraordinary Fact The work done depends on the initial and final states and the path taken between these states. BUT, the quantity Q - W does not depend on the path taken; it depends only on the initial and final states. Only Q - W has this property. Q, W, Q + W, Q - 2W, etc. do not. So we give Q - W a name: the internal energy.

The First Law of Thermodynamics (FLT) -- Heat and work are forms of energy transfer and energy is conserved. U = Q + Won change in total internal energy heat added to system work done on the system State Function Process Functions or U = Q - Wby

1st Law of Thermodynamics statement of energy conservation for a thermodynamic system internal energy U is a state variable W, Q process dependent

The First Law of Thermodynamics What this means: The internal energy of a system tends to increase if energy is added via heat (Q) and decrease via work (W) done by the system. . . . and increase via work (W) done on the system.

Isoprocesses apply 1st law of thermodynamics to closed system of an ideal gas isoprocess is one in which one of the thermodynamic (state) variables are kept constant use pV diagram to visualise process

Isobaric Process process in which pressure is kept constant

Isochoric Process process in which volume is kept constant

Isothermal Process process in which temperature is held constant

Thermodynamic processes of an ideal gas ( FLT: DU = Q - Wby ) Isochoric (constant volume) V p 1 2 Q Temperature changes FLT: Isobaric (constant pressure) V p 1 2 FLT: Q Temperature and volume change

Isothermal (constant temperature) ( FLT: DU = Q - Wby ) Isothermal (constant temperature) Q Thermal Reservoir T Volume and pressure change p V 1 2 FLT: