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USSC2001 Energy Lecture 3 Thermodynamics of Heat. Wayne M. Lawton Department of Mathematics National University of Singapore 2 Science Drive 2 Singapore 117543. Email [email protected] http://www.math.nus.edu.sg/~matwml/courses/Undergraduate/USC/2007/USC2001/ Tel (65) 6516-2749. 1.

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Ussc2001 energy lecture 3 thermodynamics of heat

USSC2001 Energy Lecture 3 Thermodynamics of Heat

Wayne M. Lawton

Department of Mathematics

National University of Singapore

2 Science Drive 2

Singapore 117543

Email [email protected]

http://www.math.nus.edu.sg/~matwml/courses/Undergraduate/USC/2007/USC2001/

Tel (65) 6516-2749

1


PRESSURE

is force per unit area and measured in Pascal’s

http://www.infoplease.com/ce6/sci/A0837767.html

Pascal's law : (päskälz') [key] [for Blaise Pascal], states that pressure applied to a confined fluid at any point is transmitted undiminished throughout the fluid in all directions and acts upon every part of the confining vessel at right angles to its interior surfaces and equally upon equal areas. Practical applications of the law are seen in hydraulic machines.

Standard atmospheric pressure is 101 325 Pa

2


DEFINING TEMPERATURE

The triple point of water

http://en.wikipedia.org/wiki/Triple_point

degrees Kelvin

Pascals

We define the temperature of a gas by

It is an empirical fact that T is the same for any two gases that are in thermal equilibrium with each other.

3


CONSTANT-VOLUME GAS THERMOMETER

The ingenious mercury thermometer shown

below can measure T at constant volume

Questions How can constant volume be maintained at different temperatures? How can density be measured?

Gas-filled bulb

Reservoir that can be raised and lowered

4


THE IDEAL GAS LAW

Amadeo Avogado 1776-1856 suggested that all

gases contained the same number of molecules

for a fixed volume, pressure and temperature

# molecules =

# moles =

where

= # molecules in a mole

= the Boltzmann constant

= the gas constant

5


COLLISION WITH A WALL

For an elastic collision between a molecule and a wall

unit

normal

vector

so the formula on page 13 of Lecture 1

where the subscript n denotes the normal components.

 1 collision changes wall momentum by

6


COLLISION RATE

of an object with horizontal velocity component

on an area A wall in a length L cylinder

unit

vector

perpen-

dicular

to wall

is

since it travels 2 L distance between

collisions alternating between the left and right walls.

Therefore

7


MOMENTUM TRANSFER RATE

Since 1 collision transfers momentum

wall

the momentum transfer rate for 1 object is

and the momentum transfer rate for all particles is

with

8


PRESSURE

Since momentum transfer rate = force,

wall

gas is a fluid, and Pascal’s law implies that the force

of a fluid is normal to a surface, the pressure

(pressure is not a vector)

The unit of pressure is

9


EQUIPARTITION OF KINETIC ENERGY

Our discussion about pressure ignored collisions.

since the directions of the particles after collision are

very sensitive to the direction between their centers at

the time of contact, the directions are random, if x,y,z

are orthogonal coordinates with x horizontal then

10


KINETIC THEORY OF GASES

Combining equations

where N is the number of particles, with equations

and

gives

Combining with the ideal gas law

gives

11


insulation

lead shot

WORK W AND HEAT Q

W

pressure

state

diagram

Q

volume

thermal reservoir

W (and Q) depend on the thermodynamic process, described by a path, not only on initial&final states

12


THERMODYNAMIC PROCESSES AND LAWS

Question Compute W for constant p and constant T

1st Law: There exists an internal energy function

such that during any

thermodynamic process

2nd Law: There exists an entropy function

such that during any thermodynamic process

13


TUTORIAL 3

1. Derive the relationship between the k and R on p 5.

2. Show that the pressure difference between heights

3. Use this pressure difference equation to show

that a container of gas having mass m weights mg.

14


TUTORIAL 3

4. Use the ideal gas law to compute the air pressure as a function of height above the ground. Assume that g is constant for this problem.

5. On p 13 show that if the gas expands by dV then E_int decreases by P dV. Do this by analysing the collisions of the molecules against the top wall of the container – which moves by a constant speed over some interval of time.

15


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