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Chapter One. Homework: None. State of a System: the Most Important

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Chapter One. Homework: None. State of a System: the Most Important August 23, 2001 Reading: Chapter One. Homework: None. State of a system: The most important concept in thermodynamics is the “state” of a system, which has to be uniquely defined by independent state variables. Any system contains matter made up of particles (atoms, ions, and molecules). The microscopic variables, such as masses, positions, and velocities of each and every particle at any given time define the microscopic state of the system. Therefore, if we know the microscopic state of a system, we can determine all of the properties of the system. However, it is impossible to know such detailed information, because there are ~1023 particles in constant motion. So, alternatively, in classical thermodynamics, we determined the properties of a system by define the macroscopic state of a system that is represented by the macroscopic variables, temperature (T), pressure (P), and volume (V), which are experimentally measurable. Equation of State: For a simple one-component system with a single phase, the state of a system can be uniquely defined by choosing any two independent variables. If the chosen two variables are fixed, then the state of the system is fixed, and hence all the other variables (properties) must have unique values corresponding to this state. Therefore, there must be a unique relationship between the different variables, which are called equation of state: V=V(T,P), or P=P(V,T), or T=T(P,V). State function: The equation of state defines the relationship between variables that are state functions; they depend on only the state values of parameters but not on the history of how the state was reached, i.e., they are path-independent. Consider a process to move a system from state 1 with volume V1(T1,P1) to state 2 with V2(T2,P2). The state of the system is represented by two independent variables T and p, and this process can proceed along an infinite number of paths, with different ways of varying T and P. However, the change of volume is always ∆V=V2-V1. So, the change of volume depends only on the volume at state 1 and the volume at state 2, independent of path, because V=(T,P) is a state function. The differential of a sate function, representing an infinitesimal change of the system is an exact differential, because state function is path independent. So, ∂V ∂V dV = dP + dT ∂ ∂ P T T P is an exact differential. There can be many other state functions. One such example is the internal energy, U. Since by defining any two variables of the system, e.g., V and T, the state of the system is defined, so also are all other state functions. Therefore, the internal energy can be given by U=U(V,T), or alternatively, U=(P,T) or U=U(P,V). Similarly the differential of internal energy is exact: ∂U ∂U dU = dV + dT . ∂ ∂ V T T V Process function: Process functions are those that do depend on the path chosen. These have to do with the interaction of the system with the surroundings. The two most important process functions are heat and work: heat adsorbed (or released) by the system and work done by (or on) the system, from the surroundings. Consider a system goes from (P1,V1,T1) to (P ,V ,T ). There are no unique values of heat (P ,V ) 2 2 2 2 2 (q) and work (w) associated with the change Path 1 of the system. The figure on left demonstrates P the case for work (w): Path 2 P2 ,V2 w = ∫ PdV = area under the P-V curve, is P ,V (P1,V1) 1 1 clearly dependent on the path chosen. So, the V work is a process function. The differential of a process function is NOT an exact differential, because the process function is path dependent, so it can’t uniquely represent the infinitesimal change of the system. We use δw and δq (istead of dw and dq) to express the differential of w and q; they are known as Pfaffian differentials. In other words, if we choose two independent variables to represent the state of a system, a state function is a unique function of these two variables, while a process function is not. Therefore, the differential of a state function with respect to these two variables is exact, while the differential of a process function with respect to these two variables is NOT but a Pfaffian differential: P = P(V,T), dP is an exact differential; V = V(P,T), dV is an exact differential; U = U(V,T), dU is an exact differential; but w ≠ w(V,P), δw is a Pfaffian differential; q ≠ q(V,T), δq is a Pfaffian differential. Extensive vs. Intensive properties: An extensive property depends on the size of the system, such as volume. An intensive property is independent of the size of the system, such as temperature and pressure. Types of sytems: (1) Isolated: does not exchange heat, work, and matter with the surroundings. (2) Closed: does not exchange matter with the surroundings. (3) Closed-Adiabatic: doest not exchange matter and heat with the surroundings. (4) Closed-Rigid: doest note exchange matter and work with the surroundings. (5) Open: exchange matter, heat, and work with the surroundings. Absolute zero temperature: Consider an one-component system, V = V (P,T) ∂V ∂V dV = dP + dT ∂ ∂ P T T P dV 1 ∂V 1 ∂V = dP + dT ∂ ∂ V V P T V T P dV = −βdP +αdT V 1 ∂V where, α = ------ Isobaric expansivity; ∂ V T P 1 ∂V β = − ------ Isothermal compressibility. ∂ V P T In 1802 GayLussac observed that the thermal expansion coefficient, α for gas is a constant. So, 1 ∂V α = , where V0 is the volume at T = 0oC and P is kept constant. ∂ V0 T P ∂V Thus, V (P,T ) = V + (T − 0) = V +αV T . 0 ∂ 0 0 T P He obtained a value of 1/267 for α. This was later refined by Regnalt in 1847 to be 1/273. (For idea gas, α=1/273.15.) V0 So, V (P,T ) = V + T . Note V0 is a function of P. 0 273.15 The lowest value of V(P,T) is zero. Consequently, the lowest possible temperature is T=-273.15oC, i.e., the absolute zero (0 K). Idea gas: A hypothetical gas that obeys Boyles’s and Charles’ laws exactly at all temperatures and pressures is called ideal gas. Usually, gases at very low pressures can be modeled as an idea gas. = Boyle’s law: P0V (T, P0 ) PV (T, P) ; V (P ,T ) V (P ,T ) Charles’ law: 0 0 = 0 ; T0 T PV P V Idea gas law: = 0 0 = const. T T0 Where P0 = standard pressure (1 atm) o T0 = standard temperature (absolute scale with 0 C=-273.15 K) V0 = V(P0,T0) = 22.414 liters Then, PV = RT, P V R = 0 0 = 0.082057 liter atm/degreee mole = 8.3144 joules/degree mole. T0.
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