# ENERGY CONSERVATION The First Law of Thermodynamics and the Work/Kinetic-Energy Theorem

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1 PH-211 A. La Rosa ENERGY CONSERVATION The irst Law of Thermodynamics and the Work/Kinetic-Energy Theorem ENERGY TRANSER of ENERGY Heat-transfer Q Macroscopic external Work W done on a system ENERGY CONSERVATION LAW The work/kinetic-energy theorem Case: inelastic collision Generalization of the work/kinetic-energy theorem undamental Energy Conservation Law Inelastic collision (revisited) Case: Pure Thermodynamics The irst Law of Thermodynamics ENERGY The total energy of a system has two distinct contributions: R CM U internal = V CM i 1 miu 2 i 2 i, j 1 P 2 ij Disordered energy (gas molecules inside the container.) u i = velocities P ij = potential energies ig. 1 E macro = (1/2) Mv CM (1/2) I 2 + Mgz Ordered energy (of the can cylinder) 1

2 A. MACROSCOPIC COMPONENT ( Ordered energy. ) The total mechanical energy of the system, associated with the macroscopic position and motion of the system as a whole. This mechanical energy comprises: i) Translational kinetic energy of the center of mass (CM) + + the rotational kinetic energy calculated with respect to the CM. ii) Potential energies associated to the position of the center of mass (gravitational potential energy, electrical potential energy, potential energy associated to the spring force, etc.) B. MICROSCOPIC COMPONENT ( Disordered energy. ) The other contribution to the energy is a vast collection of microscopic energies, known collectively as the internal energy U of the system. U comprises: The sum of individual kinetic and potential energies associated with the motion of, and interactions between, the individual particles (atoms and/or molecules) that constitute the system. These interactions involve complicated potential energy functions on a microscopic distance scale. In principle, after an appropriate choice of the zeros of the potential energy functions, one can talk about a definite value of U of the system (when the latter is in a state of thermodynamic equilibrium.) But such calculation of U can be a complicated endeavor. It is relatively simpler to calculate the changes of U. U: When a system changes its state of thermodynamic equilibrium, it is only the changes in the internal energy U that are physically significantly. 2

3 TRANSER of ENERGY Different systems can transfer energy among themselves by two processes: (1) Via heat-transfer, driven by temperature differences (2) Via work, driven by external macroscopic forces We will see that, Heat-transfer to a system is fundamentally a microscopic mechanism for transferring energy to a system. Work done on a system is a macroscopic mechanism for transferring energy to a system Heat-transfer Q It can occur via conduction, convection, and radiation The mechanism is fundamentally microscopic (at the atomic and molecular level.) heat transfer is accomplished by random molecular collisions and other molecular interactions. The direction is from the higher to lower temperature (an aspect better explained in the context of the second law of thermodynamics.) Warning: Do not confuse heat-transfer Q with the internal energy U. Heat transfer is not a property of the state of a system (a system in thermal equilibrium does not have an amount of heat or heat-transfer.) On the other hand, a system in thermal equilibrium does have (in principle) a specific internal energy. That is, Q is not a state variable U is a state variable 3

4 Macroscopic external Work W done on a system The macroscopic external work W done on a system can cause a change in either the internal energy U of the system, or the total mechanical energy E of the system Example where the external work causes a change of purely internal energy Insulation (no heat transfer Q=0) External non conservative force) ig. 2 Gas enclosed in an insulating container. The insulated walls ensure an absence of heat transfer from or toward the system (the gas.) Movable piston allows an external agent to compress the gas (by pushing the piston), thus doing work on the system. The work on the gas by the external agent results in an increase of the gas temperature (indicative of an increase in the internal energy U.) On the other hand, simply lifting the gas container (described above) would be an example of increasing the mechanical energy of the system, without changing the internal energy. 4

5 ENERGY CONSERVATION LAW The work/kinetic-energy theorem We are already familiar with the work/kinetic-energy theorem, which establishes the source (work) that causes a change in the kinetic energy of a system. We illustrated this theorem for the case of an individual particle, as well as for a system of particles constituting a rigid body. The later allowed us to solve, in a very straightforward manner, problems involving bodies rolling down an inclined plane, for example.[ But cases involving work done by internal forces in non-rigid bodies were not considered. We will encounter such cases in this section.] Case: Inelastic collision In what follows, we illustrate the need for generalizing the work/kinetic-energy theorem, in order to include cases in which disordered (microscopic) energy is involved. To that effect, let s start consider an inelastic collision. Before the collision Both particles initially at the same temperature and in thermal equilibrium v At rest m M Kinetic energy: K before = ½ m v 2 frictionless i X After the collision V ( m + M ) frictionless i Kinetic energy K after = ½ (m +M) V 2 X 5

6 Since the linear momentum is conserved mv = ( m + M ) V K after = ½ (m +M) [ ( m / (m + M) v ] 2 K after = ½ [ ( m 2 / (m + M) v 2 ] = ½ m v 2 [ ( m / (m + M) ] The change in kinetic energy is given by, K = K after - K before = [( m / (m + M) -1 ] ½ m v 2 = - [( M / (m + M) ] ½ m v 2 that is, the kinetic energy is less after the collision than before. According to the work/kinetic-energy theorem this change should have resulted from the work done by the forces acting on the system. But notice, all the external forces acting on the system (normal forces and weight) are perpendicular to the displacement of the particles, hence, their work on the system is zero (W N = 0, W W = 0.) N m N M frictionless i W 1 W 2 X Apparently, then, the work/kinetic-energy theorem W total = K appears not to be valid here. 6

7 The explanation lies in the fact that we are not including the work done by the internal friction forces. Such forces act during the inelastic collision. We say then, W internal-friction = K Thus, in this particular example, we identify the decrease in the kinetic energy in the negative work done by the microscopic internal forces. We would like to highlight that the change in kinetic energy K may include not only the macroscopic kinetic energy (of the center of mass) but also (presumably) an increase also of the microscopic kinetic energy; that is, W internal-friction = K macroscopic + K microscopic (case for the inelastic collision depicted in the figure above) The work/kinetic-energy theorem While we can in principle understand what is going on in the particular example of inelastic collision (where the system under study does not receive external heat-transfer), we would like to explore reformulating the work/kinetic-energy such that include also cases where thermal interaction (heat transfer) from the surrounding environment is allowed. As a firs step, let s express the work/kinetic energy theorem as follows, W internal + W external-non-conservative + W external-conservative = = K CM + K microscopic (generalization of the work/kinetic-energy theorem) Here W external-non-conservative refers to the work done by forces like the one pushing the piston in ig. 2 above. W external-conservative could be the work done, for example, the gravitational force. 7

8 That is, we are explicitly separating out the macroscopic work (done by external macroscopic forces, conservative and nonconservative) from the work done by microscopic forces. Similarly, we assume also that the kinetic energy changes in both macroscopic (the CM kinetic energy) and microscopic forms i) or the conservative forces component, the work can be derived from a potential energy function E p, W external-conservative = - E p which gives, W internal + W external-non-conservative + (- E p ) = = K CM + K microscopic W internal + W external-non-conservative = K CM + E p ) + K microscopic Calling K CM + E p E macro the macroscopic mechanical energy, the work/kinetic-theorem can be written as, W internal + W external-non-conservative = E macro + K microscopic ii) We can envision that, ultimately, W internal causes a change in microscopic potential energies of the interacting microscopic particles that constitute the system. That is, W internal = - P ij. i j Hence, j i - P ij + W external-non-conservative = E macro + K microscopic 8

9 W external-non-conservative = E macro + K microscopic + i j P ij Change in macroscopic mechanical energy Change in the internal energy U The last two terms in the right side of the expression above constitute what we called, at the beginning of this section, the disordered Internal Energy U of the system. Through the derivation process followed above, we notice that the work energy is deposited (transformed) into the system as either, macroscopic mechanical energy, or internal energy. The work W external-conservative done by conservative macroscopic external forces has been assimilated into the mechanical energy, while the work W internal done by microscopic forces ended up being grouped into the internal energy term. The expression above also shows that the work energy W external-non-conservative done by external non-conservative forces could end up either as macroscopic mechanical energy or internal energy (that the latter case can occur was illustrated in the example above where a gas was compressed by a piston; the force acting on the piston was the nonconservative force.) Generalization of the work/kinetic-energy theorem As illustrative as the expression above might be, it also reveals its limitations for dealing with cases in which the system is in thermal contact with a body at different temperature. Indeed, in such a case, the system can also receive energy via heat-transfer, a 9

10 process driven by temperature differences.) Accordingly the expression above needs to be modified or generalized. Q + W external-non-conservative = E macro + K microscopic + i j P ij Heat-transfer into the system In a simplified form Change in macroscopic mechanical energy Change in the internal energy U Q + W external-non-conservative = E macro + U Heat-transfer into the system caused by temperature difference Work done on the system by a nonconservative macroscopic external force Change in macroscopic mechanical energy of the system Change in the internal energy U of the system which constitutes our undamental Energy Conservation Law. Inelastic collision (revisited) Both particles initially at the same temperature and in thermal equilibrium v m At rest M frictionless i Kinetic energy K before = ½ m v 2 X After the collision V ( m + M ) frictionless i X 10

11 Here Q id the flow of energy by heat transfer, caused by temperature differences. In our case is zero.) W is the work done by external forces. In our case it is zero. U change in the internal energy E change in the mechanical energy In our case E = - [( M / (m + M) ] ½ m v 2 Accordingly, = - [( M / (m + M) ] ½ m v 2 ) + U which gives, U = [( M / (m + M) ] ½ m v 2 ) That is, the missing (ordered) kinetic energy appears as an increase in the internal (disordered)s energy U of the system. (The increase in the internal energy of the system typically manifest itself in an increase in the temperature of the system. As the temperature of the system increases above the ambient environment because of the increase in the internal energy, heattransfer subsequently occurs from the system to the environment until the system-ambient reach a common temperature. Case: Pure Thermodynamics In pure thermodynamics, one typically considers only systems whose total mechanical energy does not change, E macro = 0. The general statement of the energy conservation becomes, Q + W external-non-conservative = U 11

12 Notice Before After Work done by the external force > 0 W external-non-conservative > 0 Before After Work done by the external force < 0 W external-non-conservative < 0 It is typical to consider the work done by the system (no the work done on the system by the external non-conservative forces.) Since, according to the Newton s third law, the force exerted by the system is equal in magnitude but opposite in direction, then W external non-conservative = - W done-by-the-system Thus, for pure thermodynamic systems Q Heat-transfer-into-the-system - W done-by-the-system = U When all the terminology is understood, the subscripts are omitted and one simply writes Q - W = U The irst Law of Thermodynamics 12

13 Notice Before After Work done by the gas < 0 W < 0 Before After Work done by the gas > 0 W > 0 Question: wedge Before After W =? 13

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