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

2. PRESSURE 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.keyPascal 2 is force per unit area and measured in Pascal’s Standard atmospheric pressure is 101 325 Pa

3. DEFINING TEMPERATURE The triple point of water http://en.wikipedia.org/wiki/Triple_point degrees Kelvin 3 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.

4. CONSTANT-VOLUME GAS THERMOMETER The ingenious mercury thermometer shown below can measure T at constant volume Gas- filled bulb Reservoir that can be raised and lowered 4 Questions How can constant volume be maintained at different temperatures? How can density be measured?

5. 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 # moles = where= # molecules in a mole 5 = the gas constant # molecules = = the Boltzmann constant

6. COLLISION WITH A WALL For an elastic collision between a molecule and a wall  so the formula on page 13 of Lecture 1 unit normal vector where the subscript n denotes the normal components. 6  1 collision changes wall momentum by

7. COLLISION RATE of an object with horizontal velocity component is since it travels 2 L distance between to wall collisions alternating between the left and right walls. 7 Therefore on an area A wall in a length L cylinder unit perpen- dicular vector

8. MOMENTUM TRANSFER RATE Since 1 collision transfers momentum wall the momentum transfer rate for 1 object is 8 and the momentum transfer rate for all particles is with

9. 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 9 (pressure is not a vector) The unit of pressure is

10. 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

11. KINETIC THEORY OF GASES Combining equations and gives Combining with the ideal gas law where N is the number of particles, with equations gives 11

12. WORK W AND HEAT Q thermal reservoir insulationinsulation W Q lead shot volume pressure W (and Q) depend on the thermodynamic process, described by a path, not only on initial&final states state diagram 12

13. THERMODYNAMIC PROCESSES AND LAWS Question Compute W for constant p and constant T 13 1 st Law: There exists an internal energy function such that during any thermodynamic process 2 nd Law: There exists an entropy function such that during any thermodynamic process

14. TUTORIAL 3 2. Show that the pressure difference between heights 14 3. Use this pressure difference equation to show that a container of gas having mass m weights mg. 1. Derive the relationship between the k and R on p 5.

15. TUTORIAL 3 15 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. 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.