METR 241325 February 2002

ThermodynamicsThermodynamics

III III

Review

Hydrostatic balance

Pressure decreases exponentially with height,isothermal atmosphere:

Zeroth law of thermodynamics: thermal equilibrium

First law of thermodynamics: conservation of energy

Heat, ΔQ, is a measure of energy transfer by means of temperature differencesSpecific heat of air at constant pressure

gdz

dp

g

TRHHzp

TR

gzpzp d

d

),/exp()exp()( 00

constpp

Tm

ΔQc

Conservation of Energy

WorkWork is a measure of energy transfer by “mechanical” means. Work is energy transfer from one system to another in such a way that a difference in temperature is not directly involved.

Work = force x distance

From experiments, James Joule showed the equivalence of using mechanical energy to raise the temperature of a system and adding heat to raise the temperature

Joule’s experimental results showed that:4187 J of mechanical energy will raise the temperature of 1 kg of water from 14.5 to 15.5°C

Conservation of Energy

To determine the work done to expand a gas: dW = F dx

Assume the gas expands slowly (i.e. a reversible process)dW = p dV

If we are dealing with a unit mass of material, then volume V is replaced by specific volume α and work done on that unit mass is given by

dw = p dα, specific volume α = V/m = 1/ρ

Work is defined to be negative when work is done by the system, and positive when work is done to the system

Conservation of Energy

We can now write the First Law of Thermodynamics in a quantitative form

dU = dQ + dW

The internal energy of a substance can be changed (dU) either through heating (dQ) or through work (dW). (Energy must be added or subtracted!)

dU is a function only of the initial and final states of the substance, and is independent of the manner by which the substance changes between these two states.

Conservation of Energy

We can write the First Law in a way to describe the change in temperature of an air parcel of unit mass when heat is added and when work is done by the air parcel:

For our expanding air parcel,du = dq – dw or dq = du + dw

From earlier, using the specific heat at constant volume,du = cv ΔT and dw = p Δα

so dq = cv ΔT + p Δα

Conservation of Energy

The Ideal Gas Law, p = ρ Rd T, can be written p α = Rd T.Taking differentials gives

p dα + α dp = Rd dT

Substituting into the previous equation givesdq = cv ΔT + p Δα = cv ΔT + Rd ΔT - α Δp

or dq = cp ΔT - α Δp = cp ΔT – Δp/ρ , since α = 1/ρ

where cp ΔT represents change in internal energy and Δp/ρ represents the work done associated with an air parcel expanding or contracting due to pressure changes

Conservation of Energy

If we assume that there is no external energy input to the system (the air parcel is an isolated system), then dq = 0 = cp ΔT - α Δp

As an air parcel rises, work is done by the pressure force when the parcel expands as the pressure decreases. Hence, the temperature of the parcel decreases.

ΔT = α Δp/cp

This situation, when an air parcel moves in the atmosphere with no external energy input and no change of phase of water (no latent heat release), is called an adiabatic process.A diabatic process is when external energy is gained or lost by the parcel, such as due to radiation or latent heat release.

Conservation of Energy

Now, let us consider the adiabatic motion of an air parcel rising or sinking in an atmosphere in hydrostatic balance.

Δp = -ρ g Δz and ΔT = Δp/(ρcp)

gives ΔT = - g Δz / cp or

For adiabatic motion, the temperature decreases with height at a rate of about 9.8°C/km.This is called the adiabatic lapse rate

pcdz

dT g

padiabaticadiabatic c

g

dz

dT ,

Second Law

The second law of thermodynamics

The entropy of a closed system shall never decrease, and shall increase whenever possible.

So what is entropy and what does this mean??

The second law of thermodynamics is about “tendency” .It is a general principle that places constraints upon the direction of heat transfer and the attainable efficiencies of heat “engines”. In so doing, it goes beyond the limitations imposed by the first law of thermodynamics.

Second Law

There are many ways in which the first law (conservation of energy) can be satisfied:

cool warm

colder warmer

However, we know that this doesn’t tend to happen in nature.

Second Law

The second law of thermodynamics tells us that heat never passes spontaneously from a colder body to a warmer body.There is a “tendency” for energy transfer to only occur in one direction.

Energy spontaneously tends to flow only from being concentrated in one place to becoming diffused or dispersed and spread out.

Mathematically, we typically write the second law in terms of “entropy”

dS = dQ T

Second Law

Entropy measures the tendency of ENERGY to spread out, to diffuse, to become less concentrated in one physical location or one energetic state. Entropy is sometimes referred to as a measure of the “disorder” of a system

Order = low entropyLots of disorder = high entropy

In any isolated system, entropy will remain the same or increase

Second Law

Entropy increases when:• heat flows from a hot object to a cold object• a gas flows from high pressure to lower pressure• ice melts• water evaporates

Locally, entropy may temporarily decreasee.g. water freezing

However, entropy in the universe always increases

Second Law

Summary

Zeroth Law First Law Second LawWhen each of two systems is in equilibrium with a third, the first two systems must be in equilibrium with each other. This shared property of equilibrium is the temperature.The concept of temperature is based on this Zeroth Law

Because energy cannot be created or destroyed, the amount of heat transferred into a system plus the amount of work done on the system must result in a corresponding increase of internal energy in the system. Heat and work are mechanisms by which systems exchange energy with one another

Entropy of an isolated system can never decrease. Therefore, when an isolated system achieves a configuration of maximum entropy, it can no longer undergo change (it has reached equilibrium).