GENERAL PHYSICS II Electromagnetism & Thermal Physics 4/29/2008 Chapter XV The First Law of Thermodynamics §1 Heat, work and paths of a thermodynamic process §2 The first law of thermodynamics §3 Kinds of thermodynamic processes §4 Thermodynamic processes for an ideal gas 4/29/2008  We knew that the concepts of mechanical work and energy play an important role in studying mechanical phenomena  Concerning to thermal phenomena, there exits a new form of energy called “heat”: Heat can be transferred from one to other systems For a system with volume held constant, the effect of heat is to change the temparature of a system In general cases, for a system there exist, at the same time, transfer or exchange of heat and mechanical work → the GOAL of thermodynamics is the study of the relationships involving heat, mechanical work, the laws that govern energy transfers 4/29/2008 §1 Thermodynamic systems and processes: 1.1 Thermodynamic systems, heat and work:  In any study of heat, work transfer we must define exactly what are the objects under consideration: A thermodynamic system is any collection of objects that is regarded as a unit and that may have the potential to exchange energy with other bodies beside the system All the other bodies which have energy exchanges with the considered system are called surroundings or environment 4/29/2008  Then we must fix the convention on the symbol for heat and work: surroundings system We will always denote  by Q the quantity of heat added to the system  by W the mechanical work done by the system Therefore Q and W are understood as algebraic values, they can be positive, negative or zero Q>0 surroundings system surroundings system W>0 Work is done by the system 4/29/2008 surroundings Q , and when the gas is compressed, V2 < V1 → Wby < (it means that the surroundings does work on the gas)  In a p-V diagram, the equilibrium intermediate states are represented by the points on a curve, and the work is represented as the area under the curve p V1 4/29/2008 V2 V 1.3 Paths between thermodynamic states:  When a thermodynamic system changes from an initial state to a final state, it passes through a series of (equilibrium) intermediate states However, with the same initial and final states, the system can pass in very different ways On a P-V diagram, every way corresponds to a curve which is called the path between thermodynamic states  Examples: Two different paths between the states and : p p 1 p1 p1 p2 V V2 → : keep the pressure constant at p1 while the gas expands to the volume V → : reduce the pressure to p2 at constant volume V2 4/29/2008 p2 V V1 → 4: reduce the pressure at the constant volume V1 → 2: keep the pressure constant at p2 while the gas expands to the volume V2  It is important to remark that with the same intial and final states:  The work done by the system depends on the intermediate states, that is, on the path,  Like work, the heat which the system exchanges with the surroundings depends also on the path Examples: p p 2 V In an isothermal expansion of the gas we must supply an input heat to keep constant temperature 4/29/2008 V Gas can expand in an container which is isolated from surroundings (no heat input) §2 The first law of thermodynamics: 2.1 Internal energy of a system:  The internal energy of a system is the energy that the system owns We can define: Internal energy = ∑kinetic energies of constituent particles + ∑potential energies between them (Note that the internal energy does not include potential energy arising from the interaction between the system and its surroundings, for example, system and gravitaitonal field)  For an ideal gas we know how can calculate the internal energy But for any real system, the calculation of the internal energies by this way would be very complicated 4/29/2008 10 §3 Kinds of thermodynamic processes: We know that there are many different paths between thermodynamic states We will study four specific kinds of thermodynamic processes which are important in practical applications 3.1 Adiabatic process:  Definition: Adiabatic process is defined as one with no heat transfer into or out of a system, Q = V  Examples: Gas in a container which is surrounded by a thermally isolating material A expansion (or compression) of gas which takes place so quickly that there is not enough time for heat transfer  From the 1st law: Δ = U2 – U1 = - W U (adiabatic process) p 4/29/2008 13 3.2 Isochoric process:  Definition: This is a constant-volume process  Example: A gas in a closed constant-volume container p  When the volume of a thermodynamic system is constant, it does no work on its surroundings V W=0  From the 1st law: Δ = U2 – U1 = Q U (isochoric process)  Since the system does no work → all the energy (heat) added remains in the system → the iternal energy increases 4/29/2008 14 3.3 Isobaric process:  Definition: This is a constant-pressure process  In a isobaric process, none of three quantities Δ Q, W is zero U, p V  Work done by the system is easily calculated: W = p (V2 – V1 ) 4/29/2008 15 3.4 Isothermal process:  Definition: This is a constant-temperature process p  To keep temprature constant, the system must exchange heat with the surroundings, and the exchange must be slowly that thermal equilibrium is maintained V  In general, in a isothermal process, none of Δ Q, W is zero U,  Only in the case of an ideal gas, the internal energy U ~ T → Δ = in a isothermal process U 4/29/2008 16 §4 Thermodynamic processes for an ideal gas:  In this section, by applying the 1st law of thermodynamics we study in more details thermodynamic processes for an ideal gas  For an ideal gas, with the help of kinetic-molecular model, we know that the internal energy of an ideal gas depends only on its temperature, not on its pressure or volume  Owing to the explicit relation between the internal energy U and temperature T we can find explicit equations which relate heat, work and internal energy 4/29/2008 17 4.1 Constant-volume and constant-pressure heat capacities of an ideal gas:  We knew the concept of heat capacity of an ideal gas in a constant-volume process Now consider more general cases of thermodynamic process  The general definition of heat capacity is the following equation: where Δ is the quantity of heat added to the system for increase Q Δ in temperature T This definition can give rise different heat capacities which depend on the paths of thermodynamic process 4/29/2008 18  The constant-volume heat capacity is defined by Notes:  Here we replace Δ by Δ because no work done in the process Q U  If we understand CV as molar constant-volume heat capacity, then Δ is the heat added per mole Q  The constant-pressure heat capacity: For a constant-pressure process the effect of the heat added to the system is twofold: to increase the internal energy and to work 4/29/2008 19 • Applying the 1st law we can write • At the limit Δ → : T • In the case of an ideal gas, U depends only on T , we have • Using the equation of state of an ideal gas we obtain the relation for the molar heat capacities CP and CV : CP = CV + R (See experimantal values of CV and CP given in textbook, p 740, tab 19.1) 4/29/2008 20 4.2 Isothermal processes for an ideal gas:  For a fix amount of ideal gas, from the state equation thermal processes are represented by P-V curves shown in the picture These curves are called isotherms  Calculate the work done by the gas in the isothermal expansion A → B with a fixed temperature T = T0 (see the picture): where K = n R, n is the number of moles 4/29/2008 21  For the case of an ideal, the internal energy U depends only on temperature → Δ = in isothermal processes By the st law, U the heat added to the gas for keeping constant temperature equals to the work done by the gas: Q = W 4.3 Adiabatic processes for an ideal gas:  According to the definition, in an adiabatic process no heat transfer takes place → Q =  Remember for an ideal gas, U depends only on temperature T , and → for an adiabatic expansion (or compression) of gas, from 1st law we have Q = Δ + P Δ = CV Δ + P Δ = U V T V 4/29/2008 22 For n moles of ideal gas n CV Δ = - P Δ = - n(RT/V) Δ T V V apply the state equation for n moles of ideal gas For an infinitesimal process we have R CP  V CP C     1 CV CV CV dT dV   1) (   T V 4/29/2008 dT R dV   T CV V C  P  where CV (the ratio of heat capacities) (γ is always positive ! -1) 23  By integrating we obtain ln T  1) ln V  (  constant  ln(TV 1 )  constant ln T  V   ln  constant  TV   constant It means that for an adiabatic process from a initial state (T1, V1)  to a final state (T2, V2) : T V   V 1 T  1 2  We can convert the relation for (T, V) into a relation for (p, V) PV nR by substituting T  PV  V  constant nR PV   constant It means that for an adiabatic process: 4/29/2008 P1V1 P2V 2 24  Calculate the work done by the gas From the 1st law: W    nCV (T2  )  V (T1  ) U  T nC T Using PV = n RT we have C W  V ( PV1  2V2 )  P ( PV1  2V2 ) P 1 R 1   P-V curves for adiabatic processes are shown in the picture, together with isoterms for comparison Remark:  Adiabatic curves are more sloping than isoterms (Compare PV   constant with PV  constant )  An adiabatic expansion (for example A → D) causes a drop in temperature (T → T1) 4/29/2008 • Adiabatic curves: blue dashed • Isoterms: solid black 25 Summary Δ = Q-W U  The first law of thermodynamics: The change of internal energy  The work done by a system: The quantity of heat added to the system The work done by the system V2 W  PdV V1  The relation between heat capacities: CP = CV + R CP   CV 4/29/2008 26  For an isothermal process of an ideal gas: Δ =0 U   V Q  nRT0 ln  W   V 1   For an adiabatic process of an ideal gas:  TV   constant PV   constant C W  V (T1  )  V (PV1  2V2 )  nC T P (PV1  2V2 ) P 1 R 1  4/29/2008 27 .. .Chapter XV The First Law of Thermodynamics §1 Heat, work and paths of a thermodynamic process §2 The first law of thermodynamics §3 Kinds of thermodynamic processes §4 Thermodynamic... Having the concept of the internal energy, we can formulate the first law of thermodynamics 4/29/2008 11 2.2 Formulation of the first law of thermodynamics:  Consider a change of state of the system... Q-W U  The first law of thermodynamics: The change of internal energy  The work done by a system: The quantity of heat added to the system The work done by the system V2 W  PdV V1  The relation