The fundamental equations for a closed system described by state variables T, S,
ID: 1719114 • Letter: T
Question
The fundamental equations for a closed system described by state variables T, S, P and V in terms of U, H and F:
dU = TdS – PdV
dH = TdS + VdP
dF = -SdT – PdV
They tell us how these thermodynamic potentials change with their natural variables. However, T and P are the state variables that are easiest to control in most experiments. Please derive expressions for dU, dH and dF in terms of dT and dP. The expressions should only contain quantities that are easy to measure such as V, T, P, v , T and cp.
Explanation / Answer
Central equation of thermodynamics is the fundamentalü equation of thermodynamics (FEOT). Describes change of internal energy of a system that is: o initially in one state of equilibrium o and is perturbed to another state of equilibrium. In its simplest form, FEOT is given by a change inü internal energy of a system as described by the sum of the change in heat, Q, and work, W: Heat and work
differentials are inexact since they are pathü
dU dQ dW = + 1 dependend
U TdS PdV dN YdX = + +t (depend on past history). Heat differential is described mathematically as:ü Where T and S are the temp and entropy of the system.ü Similarly, the work differential is given by:ü dQ TdS = 2 1st term on right: hydrostatic work on system (hence, theü negative sign) where P and V are the pressure & volume. dU %dV dN YdX = + + o µ 3
2nd term on right: chemical work done by the system (hence, the positive sign) where µ & N are the chemical potential and number of particles in the system. The last term allows for other forms of work performedü on or by the system. The full form of the FEOT is:
Two important points concerning the FEOT: o 1st point: FEOT Variables – differentials and their prefactors: • Differential Variables: » Called extensive or state variables. » Directly measurable quantities & describe the state of the system prior to (initial state) and following (final state) the system change. » Changes in state variables are path independent (do not depend on past history). • Pre-factor variables to the differential variables: » Called intensive or field variables. Measurable indirectly only, by means of the response of an extensive variable to a perturbed system. • Work performed on or by system = product between an extensive or state variable and an intensive or field variable
Mathematical Properties of the FEOT FEOT is an exact differential:ü o Definition of an Exact differential: In Thermodynamics, the total energy E of our system (as described by an empirical force field) is called internal energy U. U is a state function, which means, that the energy of a system depends only on the values of its parameters, e.g. T and V, and not on the path in the parameters space, which led to the actual state. This is important in order to establish a relation to the microscopic energy E as given by the force field. Here, E is completely determined by the actual coordinates and velocities, which define the microscopic state {ri, pi }.
The first law of Thermodynamics states the conservation of energy, U = const (VII.1) and we have used this in the microcanonical ensemble, where the total energy is fixed. In open systems, heat (Q) can be exchanged with the environment, and this situation is modeled in the canonical ensemble. U = Q, if the Volume V is constant. If the Volume is not if the Volume V is constant. If the Volume is not fixed, work (W) can be exchanged with the environment, dW = F dz = pdV (VII.2) The first law therefore reads:
dU = dQ + dW = dQ –pdVUsing the first law, we can rewrite the internal energy as: dU = dQ = T dS Therefore, we changed the dependence of the internal energy from U(T) to U(S). Including volume change, we can write:
dQ = T dS = dU – dW
The second law states, that the entropy production is positive, i.e. we can write: dS dQ T 0. dS is the total entropy-change and dQ/T is the reversible part. If we have dV=0, we can write: T dS dU 0. This is the condition, that a process happens spontaneously. I.e., processes will happen in a way, that the property TS-U becomes a maximum, or, vice versa, U T S = min. I.e., nature minimizes the value of U-TS in its processes
Since the total energy is conserved, the only driving force is the second law, the entropy will be maximized. dS = dS1 + dS2 0 (1 is the system, 2 the environment) The entropy change of the environment, dS2 is given by the heat exchanged, dS2 = dQ T = dU1 T (U1 is the change of internal energy of the system). Therefore, dU1 T + dS1 0 combines the 1st and 2nd law to derive the driving force for the system 1 (remember, that we have V=const. in the whole discussion!).
The driving force for complex processes is the maximization of the entropy of the total system. Since we can not handle the whole universe in all our calculations, we found a way to concentrate on our subsystem by looking at U-TS. This is called the Helmholz free energy:
F = U T S (VII.7) This is the fundamental property we are intersted in, because:
• F= F(T,V): F depends on the variables T and V, which are experimentally controllable, while U=U(S,V) depends on S and V. We do not know, how to control entropy in experiments. In particular, F is the energetic property which is measured when T and V are constant, a situation we often model in our simulations. • F = Ff Fi = Wmax is the maximum amount of work a system can release between an initial (i) and final (f) state
. In the first law dU = dQ + dW, we can substitute dQby TdS, since the latter is always large due to second law T dS dQ to get: dU T dS + dW, or: dW dU T dS = dF
Therefore, the maximal work is always greater or equal the free energy. In other words, a certain amount of internal energy dU can never be converted completely into work, a part is always lost due to entropy production. • If the system is in contact to environment, there is no more a minimum (internal) energy principle available. In principle, energy minimization as we have discussed before, does not make any sense, neither for the subsystem, not for the whole. Energy is simply conserved and we have a micro-canonical ensemble, however, including the whole universe. The free energy however, restores a minimization procedure again: Systems will evolve in order to minimize the free energy. This, however, is nothing else than entropy maximization of the whole system
Beginning with the first law of thermodynamics for an open system, this is expressed as:
where U is internal energy, Q is the heat added to the system, W is the work done by the system, and since no mass is transferred in or out of the system, both expressions involving mass flow are zero and the first law of thermodynamics for a closed system is derived. The first law of thermodynamics for a closed system states that the increase of internal energy of the system equals the amount of heat added to the system minus the work done by the system. For infinitesimal changes the first law for closed systems is stated by:
If the work is due to a volume expansion by dV at a pressure P then:
For a homogeneous system undergoing a reversible process, the second law of thermodynamics reads:
where T is the absolute temperature and S is the entropy of the system. With these relations the fundamental thermodynamic relation, used to compute changes in internal energy, is expressed as:
For a simple system, with only one type of particle (atom or molecule), a closed system amounts to a constant number of particles. However, for systems undergoing a chemical reaction, there may be all sorts of molecules being generated and destroyed by the reaction process. In this case, the fact that the system is closed is expressed by stating that the total number of each elemental atom is conserved, no matter what kind of molecule it may be a part of. Mathematically:
where Nj is the number of j-type molecules, aij is the number of atoms of element i in molecule j and bi0 is the total number of atoms of element i in the system, which remains constant, since the system is closed. There is one such equation for each element in the system
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