Temperature and Kinetic Theory

Before our current understanding of heat took shape, it was believed that objects heated up because of a fluid called “caloric”.

Antoine Lavoisier proposed caloric theory – the idea that a fluid called “caloric” would flow from hotter bodies to colder bodies causing each of them to experience a temperature change

French Chemist and Revolutionary

1743-1794

James Joule disproved caloric theory (heat as a fluid) by demonstrating a relationship between heat and work.

It was realized that heat is a form of energy. But to be that, it couldn’t be a fluid. Caloric theory was abandoned.

English Physicist and Brewer

1818-1889

Joule’s Experiment

As a substance warms up, energy is absorbed by molecules causing them to move faster

Therefore, the average kinetic energy of the molecules increases.

Most substances expand when they are warmed because of the excessive molecular movement.

∆L = αL0∆T

L = Initial length (any unit)

T = Temperature (C)

α = coefficient of linear expansion (C-1)

Three Mechanisms for Energy Transfer

This process of heat transfer through contact is called heat conduction.

Examples:Cold clothes in your closet get

placed on your warm body

Cold ice cubes are placed in a warm drink

In each case, the cold object warms up and the warm object cools down until a temperature in the middle is reached (thermal equilibrium).

Big picture – Energy flows from warm objects to cold objects. This flow of energy is called heat (Q).

Details – Molecules in the warmer object collide with molecules in the colder object and transfer their kinetic energy to them

H = Q / t = kA∆T / LH = rate of heat transfer (W or J/s)

Q = heat (J)

k = thermal conductivity

A = cross-section area (m2)

T = temperature (C)

L = length through which heat is transferred (m)

Energy can transfer due to currents caused by pressure differences.

Warmer molecules carry energy.

Process called convection

(No contact necessary)

Energy transfer can also be caused by electromagnetic waves.

Waves carry energy through space.

Process called radiation.

(No contact needed, no substance needed)

Two rooms, each a cube 4.0 m per side, share a 12-cm-thick brick wall. Because of a number of 100-W light bulbs in one room, the air is at 30 C, while in the other room it is at 10 C. How many of the 100-W bulbs are needed to maintain the temperature in the warmer room since heat is traveling through the wall?

Ideal Gas Laws

An ideal gas is one in which:

•Particles are point particles (only translational motion)

•All collisions are perfectly elastic (no KE lost)

•Particles are spaced far apart

•Particles do not attract / repel each other

•Lots of particles

http://www.cabrillo.edu/~jmccullough/Applets/OSP.html

P1V1 = P2V2

Boyle’s Law (isothermal system)

V1 / T1 = V2 / T2

Charles’ Law (isobaric system)

P1 / T1 = P2 / T2

Gay-Lussac’s Law (isovolumetric system)

P1V1 / T1 = P2V2 / T2

Combined Gas Law (number of moles constant)

PV = nRT

P = Pressure (Pa or N / m2)

V = Volume (m3)

n = number of moles

T = Temperature (K)

R = Universal Gas Constant (8.31 J / mol K)

PV = NkBT

N = number of molecules

kb = Boltzmann’s Constant (1.38 x 10-23 J / K)

Temperature is not a measurement of heat.

•Heat is energy that flows from one object to another. It is not inherent to a specific substance.

•Temperature is.

Temperature – the measurement of the average translational kinetic energy of the molecules that make up a substance

KEave = 3/2 kBT

KEave = average molecular kinetic energy (J)

kB = Boltzmann’s constant

T = temperature (K)

Since KE = ½ mv2 for the average molecule

½ mv2 = 3/2 kBT

v2 = 3kBT/m

vrms = √(3kBT/M)vrms = root mean square speed (v of average molecule)

M = molecular mass (kg)

The average kinetic energy takes into consideration the speeds of all the molecules. The speed of the average molecule is called the “root mean square speed” or vrms

Calculate the rms speed of helium atoms near the surface of the Sun at a temperature of about 6000 K.

What is the rms speed of nitrogen molecules contained in an 8.5-m3 volume at 2.1 atm if the total amount of nitrogen is 1300 mol?

A tank in a room at STP contains 26.0 kg of O2 gas at a gauge pressure of 8.70 atm. If the oxygen is replaced by helium, how many kilograms of the latter will be needed to produce a gauge pressure of 7.00 atm?

When energy (heat) flows into an object, the temperature goes up because:

• The molecules move faster

• The kinetic energy of the molecules increases

Can you have a temperature increase without adding heat?

Laws of Thermodynamics

We already know that the kinetic energy of an object is changed by doing work on the object.

Work causes objects to move faster (W = ΔKE)

The same is true for molecules…to make them move faster without heat, work has to be done on them.

When work is done on a system by compressing it…

The temperature goes up

Work can also be done by a system when it expands….

Causing its temperature to go down

The internal energy (U = molecular KE and PE) of an object depends on two things:

• How much heat goes in / comes out of system

• How much work is done on / done by the system

∆U = Q + WU = internal energy (J)

Q = heat (J)

W = work (J)

This formula represents the First Law of Thermodynamics. The law states that

A system’s internal energy is equal to the heat added and the work done on the system.

Observing what heat is doing is relatively easy…

Objects that are warming up are absorbing heat (+)

Objects that are cooling down are releasing heat (-)

Observing work being done on or by a system is a little more tricky…

Systems that are compressed have work done

on them (+)

Systems that expand do work on their surroundings

(-)

Sign Conventions Heat Work

In + +

Out - -

Since +W causes the gas to compress, it causes a decrease in volume

Since –W causes the gas to expand, it causes an increase in volume.

W = -P∆VW = work (J)

P = pressure (Pa or N / m2)

V = volume (m3)

An ideal gas is heated and expands at a constant total pressure of 3.0 atm from 400 mL to 660 mL. Heat then flows out of the gas at constant volume, and the pressure and temperature are allowed to drop until the temperature reaches its original value. Calculate (a) the total work done by the gas in the process, and (b) the total heat flow into the gas.

Type of Process

or Step

Definition Result in

First Law

Examples

Isothermal ∆T = 0, ∆U = 0 Q = - W Water boiling in teapot

Adiabatic Q = 0 ∆U = W Clouds forming in atmosphere

Isovolumetric ∆V = 0, W = 0 ∆U = Q Soda pop heating in car

Isobaric ∆P = 0 ∆U = Q + W

= Q - P∆V

Piston moving because of heated gas

Types of Thermodynamic Processes

P-V Diagrams

P-V diagrams are used to represent thermodynamic systems.

Temperature lines are hyperbolic because of PV = nRT

If a system’s number of molecules and temperature remains constant, the product of the pressure and volume must also be constant.

If piston is isothermal (∆T = 0), then as heat is added, work is done by the system (Q = - W).

Volume goes up and pressure goes down.

Different lines represent different types of systems

AD is an isovolumetric system (V is constant).

AB is an isothermal system (T is constant)

BD is an isobaric system (P is constant)

P-V diagrams that are closed represent cyclical thermodynamic systems.

Examples

Engines

Refrigerators

We know that

W = -P∆V

On a P-V diagram, work is…

the area under the curve.

Important because it tells you how efficient thermodynamic systems are (i.e. how much energy is converted to work)

Adiabatic systems are ones that have a temperature change without heat being exchanged.

The lines are NOT hyperbolic like isotherms.

Consider the following two-step process. Heat is allowed to flow out of an ideal gas at constant volume so that its pressure drops from 2.2 atm to 1.4 atm. Then the gas expands at constant pressure, from a volume of 6.8 L to 9.3 L, where the temperature reaches its original value. Calculate (a) the total work done by the gas in the process, (b) the change in internal energy of the gas in the process, and (c) the total heat flow into or out of the gas.

Heat Engines and Car Motors

Heat engines are designed to use the heat that flows from a high temperature to a low temperature.

The separation between can either be spatial (different regions) or temporal (different points in time)

As heat naturally flows from the hot region to the colder region, work is produced.

1. Cold water gets heated creating steam (Q in)

2. Steam at high pressure pushes piston creating work (W out)

3. Cold steam condenses creating water (Q out)

4. Water returned to boiler via pump to repeat process (W in)

Steam Engine

In an engine, the high temperature is achieved when a mixture of fuel and air is ignited by a spark plug.

The expanding gas does work against the piston pushing it downward.

The gas mixture left over is cooler and it is pushed out as exhaust.

The cycle repeats over and over which turns a crankshaft making the car move forward

The efficiency of an engine is defined as the ratio of the amount of work output (W) relative to the amount of heat input (QH)

e = W / QH

After work is done, there is still heat left over (QL)

QH = W + QL

Therefore,

e = (QH – QL) / QH

e = 1 – (QL / QH)

Can an engine ever be 100% efficient?

NO! Violation of Second Law of Thermodynamics which states that systems move toward a greater state of “entropy” or “randomness.” i.e., no device can transform all of its heat completely into work (Kelvin-Planck statement).

A Carnot engine is a theoretical design that gives an impression of what an “ideal” engine would look like.

Because no heat would be lost as friction, the temperature would be absolutely proportional to the heat.

Therefore, the efficiency is calculated using the internal temperatures of the hot and cold reservoirs rather than the heat flows.

ec = (TH - TL ) / TH = 1 – (TL / TH)

A heat engine utilizes a heat source at 550 C and has an ideal Carnot efficiency of 28%. To increase the ideal efficiency to 35%, what must be the temperature of the heat source?

Refrigerators and Air Conditioners

Refrigerators follow this principle. Work is done to move heat (energy) a cold area to a warmer area. Otherwise, it wouldn’t move that direction.

What is left is a cold region that has gotten even colder and a warm region that has gotten even warmer.

Can heat ever flow from cold regions to warm regions?

YES! But it has to follow the Second Law of Thermodynamics which states that in order for heat to flow backwards, work has to be done on the system (Clausius statement)

A compressor does work on refrigerant vapor making it hotter.

As it moves outside the refrigerator, it loses heat to the outside and condenses.

As it moves into the refrigerator, it expands and becomes a cooler liquid.

While in the refrigerator, it takes in heat from the refrigerator and becomes vaporized.

The compressor does work to heat it up and move it out repeating the process.

Both statements of the Second Law of Thermodynamics can be summed up this way…

Natural systems tend to move toward a greater state of disorder.

Furthermore, although energy is conserved, it becomes less useful

For example,

• A mixture with layers initially cannot be shaken with the expectation that the layers will be restored.

• A coffee cup that drops and breaks cannot reassemble itself.

• Your room?

When heat flows from hot to cold, the molecules end up all having the same temperature and kinetic energy (less order). To flow from cold to hot, work has to be done because that direction would produce more order.

Engines cannot produce the same amount of work as the initial heat because some of the heat is converted to other less useful forms (internal energy, radiant energy)