Lecture 5 The First Law and Mechanisms of Energy Transfer Last time we saw that the energy transferred by work and the energy transferred by heat both depend on the process by which a gas goes from the initial to final state However experiments show that the sum of these two energy transfers does not depend on the process This allows us to define an additional state variable for the system the total energy stored inside Eint The internal energy can be changed both by mechanical work and heat and the first law of thermodynamics states that Eint Q W This is really a statement of conservation of energy in words it means that the change in energy inside a system equals the total energy flowing into or out of the system Cyclic Systems Consider a gas that takes the following path on the P vs V graph P initial final state V In this case the initial and final states are identical Since internal energy is a state variable this means that Eint 0 Using the First Law this tells us that Q W for this process in other words the energy transferred by work is equal and opposite to the energy transferred by heat To study other applications of the First Law it helps to consider a few idealized processes Ideal Processes 1 Adiabatic A process in which no energy is transmitted by heat i e Q 0 an example would be the compression or expansion of a gas in a perfectly insulated container The first law tells us that Eint W for an adiabatic process Special case the adiabatic free expansion Before After barrier breaks Thin barrier Gas Gas Insulating wall In the adiabatic free expansion no work is done and no heat is transferred Thus Eint 0 for such a process 2 Isobaric process This is one in which the pressure is held constant Both Q and W are usually non zero but calculation of W is simplified V V f f Vi Vi W PdV P dV P V f Vi P Vi V f 3 Isovolumetric process volume is constant during the process W is 0 since work is only done when volume changes so Eint Q 4 Isothermal process temperature held constant during process Let s assume we have an ideal gas that is expanded or compressed isothermally Then Vf Vf nRT W PdV dV V Vi Vi Vf dV Vf nRT nRT lnV Vi V Vi nRT lnV f lnVi nRT lnVi lnV f Vi nRT ln Vf What about Q It turns out as we ll prove later that the internal energy of an ideal gas depends only on the temperature So for an isothermal process Eint 0 Which means according to the 1st Law Vi Q W nRT ln Vf Energy Transfer Mechanisms We ve talked about the transfer of energy into or out of a system but haven t specified how it happens There are several possible ways for energy to move from place to place it s quite possible for more than one of them to be going on at the same time The first mechanism we ll discuss is thermal conduction this is heat transfer by direct contact Hot region Fast moving molecules Cold region Slow moving molecules When a fast moving molecule collides with a slowmoving one the result is slow moving molecule gains energy fast moving molecule loses energy Energy is transmitted from the hot to the cold region Rate at which energy is transferred P depends on the temperature difference and on the material Consider a slab of material with different temperatures on each side P gets larger as Th Tc A x A gets larger x gets smaller T Th Tc gets larger We can summarize the behavior of P with the equation T P A x If we take the limit where x is very small this becomes dT dT P A or P kA dx dx where k is the thermal conductivity of the material k varies a great deal among materials Copper transmits heat about 5000x better than wood does That s why you can grab a hot metal pot by a wooden handle without getting burned Convection The movement of hot material into a cooler region or vice versa is called convection Example cool air from an air conditioner is denser than the warm air in a room so it tends to fall this means one should install A C vents near the top of a room and heating vents near the bottom so that the heat can flow upwards by convection Rate of convection depends on details of the material shape of the room etc Radiation The final method of energy transfer is via electromagnetic radiation this is the only method that can transmit energy across a vacuum Stefan s Law tells us that the rate of energy radiated from an object is P AeT 4 where is a constant 5 7 x 10 8 Wm 2K 4 A is the surface area of the object e is the emissivity and is equal to the fraction of incoming radiation absorbed by the object therefore e must be between 0 and 1 If an object is immersed in an environment that has a temperature To it will both absorb and emit energy via radation The net power radiated will be P Ae T 4 To4