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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 PRESSURE is force per unit area and measured in Pascal’s 1 pascal (Pa) 1 N m -2 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 Ttriple 273.16 K p triple 611.73 Pa degrees Kelvin Pascals We define the temperature of a gas by p(T ) T 273.16 K lim gas0,vol const p ( 273.16 K ) 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 p atm 1.01 10 Pa 3 3 mercury 13.6 10 kg / m 5 Questions How can constant volume be maintained at different temperatures? How can density be measured? p T Gasfilled bulb h p patm gh 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 = P Vol / n R T where N A 6.02 10 32 k 1.38 10 P Vol / k T 23 = # molecules in a mole J / K = the Boltzmann constant R 8.314 joules / (mole K) = the gas constant 5 COLLISION WITH A WALL For an elastic collision between a molecule and a wall unit M1 M M 2 normal 2 M1 vector V ' 1 ' V1 u n V2 V2 M1 M 2 so the formula on page 13 of Lecture 1 ' V1,n V1,n 2V2,n M1 ' V2,n 2 M 2 V1,n V2,n where the subscript n denotes the normal components. 1 collision changes wall momentum by 2 M 1 V1,n 6 COLLISION RATE of an object with horizontal velocity component on an area A wall in a length L cylinder z unit vector un y x Vol AL is Vx M V perpendicular to wall | Vx | /( 2 L) since it travels 2 L distance between collisions alternating between the left and right walls. Therefore Rcol A | Vx | / Vol 1 2 7 MOMENTUM TRANSFER RATE Since 1 collision transfers momentum 2M | Vx | u n un M V wall the momentum transfer rate for 1 object is Rmom (1) Rcol p ( AV M / Vol ) u n 2 x and the momentum transfer rate for all particles is 2 R (all ) A avg(V ) u with density mom x n 8 PRESSURE Since momentum transfer rate = force, un M V wall gas is a fluid, and Pascal’s law implies that the force of a fluid is normal to a surface, the pressure P avg(V ) 2 x (pressure is not a vector) The unit of pressure is Pascal Newton / Meter 2 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 avg(V ) avg(V ) avg(V ) avg(V ) 2 x 2 y 2 z 1 3 2 10 KINETIC THEORY OF GASES Combining equations M N / Vol where N is the number of particles, with equations P avg(V ) 2 x gives and avg(V ) avg(V ) 2 x 1 3 2 2 N 2 1 P avg ( MV ) 3 Vol 2 Combining with the ideal gas law N P Vol / k T gives 2 2 1 k T avg ( MV ) 3 2 11 i n s u l a t i o n lead shot pi W WORK W AND HEAT Q Vf W dW pdV Vi pressure pi state diagram Q thermal reservoir pf volume Vi Vf 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 Vf W p const pdV p (Vf Vi ) WT const V Vf Vi i nRT Vf dV n RT ln V Vi 1st Law: There exists an internal energy function E int E int (V , p) such that during any thermodynamic process E int Q - W 2nd Law: There exists an entropy function S S (V , p ) such that during a reversible thermodynamic process S (Vf , p f ) (Vi , p i ) Sf dQ Q T dS Si T 13 TUTORIAL 3 1. Derive the relationship between the k and R on p 5. 2. Show that the pressure difference between heights h1 and h2 is P2 P1 g (h2 h1 ). 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