By considering the volume V and entropy S as the two independent variables in the thermodynamic equation dE = TdS−PdV , derive the Maxwell relation between the derivatives ∂T/ ∂V and ∂P/ ∂S .
By considering the volume V and entropy S as the two independent variables in the thermodynamic equation dE = TdS−PdV , derive the Maxwell relation between the derivatives ∂T/ ∂V and ∂P/ ∂S .
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By considering the volume V and entropy S as the two independent variables in the thermodynamic equation dE = TdS−PdV , derive the Maxwell relation between the derivatives ∂T/ ∂V and ∂P/ ∂S .
5) a) Beginning with the fundamental relation, dE = Tds - PdV, show that S = S(E, D) b) Use the answer to part a to...
5) a) Beginning with the fundamental relation, dE = Tds - PdV, show that S = S(E, D) b) Use the answer to part a to show that: c) Legendre transform S with respect to the conjugate variables (E to obtain the Helmholtz free entropy s and show that the resulting expression for dø indicates that ° = .D Note: The expression dE = Tds - PdV is sufficient proof that E = E(S.V)
02) Consider a hydrostatic system represented by the thermodynamic variables volume V, pressure P and temperature...
02) Consider a hydrostatic system represented by the thermodynamic variables volume V, pressure P and temperature T. a) Consider entropy S = S(T, V) and derive the equation TdS. Tas = Cvat +T (1) dV. V Show that this equation can be written as follows BT TdS = CydT + PDV where Cv is the thermal capacity at constant volume, B is the isobaric expansiveness and K is isothermal compressibility: b) Consider a gas described by the equation of state...
5. (a) Derive the following Maxwell relation for a fluid system: ()-), (b) Show that for...
5. (a) Derive the following Maxwell relation for a fluid system: ()-), (b) Show that for an ideal gas at a given temperature T, the chemical potential difference between an arbitrary pressure, P, and a reference pressure, Po, is given by u(P) – u(Po) = kbT In (P/Po). (c) For the case of an incompressible liquid, the volume per particle, v = V/N, is independent of the pressure; show that in this case, one has u(P) – ”(Po) = v...
2. Take the entropy to be a function of the independent variables T and V. For...
2. Take the entropy to be a function of the independent variables T and V. For the van der Waals gas, derive an expression for the entropy as a function of volume at a given (fixed) temperature. Hint: You will want to take advantage of Maxwell's relations. Your answer will contain an undetermined constant of integration.
(Thermodynamics, Maxwell relations. Mathematics) 3 Consider the microcanonical formula for the equilibrium energy E(S, V,N) of...
(Thermodynamics, Maxwell relations. Mathematics) 3 Consider the microcanonical formula for the equilibrium energy E(S, V,N) of some general system.58 One knows that the second deriva- tives of E are symmetric; at fixed N, we get the same answer whichever order we take partial derivatives with respect to S and V (a) Use this to show the Marwell relation (3.70) OS S,N (This should take two lines of calculus or less.) Generate two other similar formulae by taking other second partial...
1. (a) Write the total differentials of s-S(V, T),S- S(P, T). Get used to it. This...
1. (a) Write the total differentials of s-S(V, T),S- S(P, T). Get used to it. This is the first step in many derivations in thermodynamics. We'll actually come back to these particular ones in the next lecture or so. (b) Now, from the 1st law we know that du dq+ dw. () Take the differential along a reversible path: dU - dgrev+ dwre. [Question: why don't we write dUrev? Yes, you must answer this and the other questions in blue.]...
A mixture of two monatomic ideal gases consists of Na molecules of gas A and Na...
A mixture of two monatomic ideal gases consists of Na molecules of gas A and Na molecules of gas B in a container of volume V at temperature T. (a) Obtain an expression for natural log of the number of the accessible microstates for each species (ie, ln S, and In Ω), (b) Show that the entropy of the mixture system is 4. EA and Ea are the total energies for the two molecular species, m, and m , the...
3. In a second-order phase transition, there is no difference in entropy or volume between the tw...
3. In a second-order phase transition, there is no difference in entropy or volume between the two phases and therefore the Clausius-Clapeyron equation cannot be applied. Two Ehrenfest equations can, however, be found. They follow from the the equality in entropy or volume across the phase transition, and they relate dp/dT to differences in α and C, or K between the two phases (a) By setting dSdS2, find dp/dT in terms of the differences in a and Cp between the...
02) Consider a hydrostatic system represented by the thermodynamic variables volume V, pressure P and temperature...
02) Consider a hydrostatic system represented by the thermodynamic variables volume V, pressure P and temperature T. A) A hydrostatic system undergoes a phase transition from first order to temperature To = 300 K and pressure Po = 1 x 100 Pa. In this case, it is verified that the variation of the specific volume of the system Av = 2 x 10 m3/Kg and the variation of the enthalpy specific Ah = 60 KJ/Kg. Determine the variation of the...
06) Suppose a paramagnetic system that is under the action of an auxiliary (magnetic) field H....
06) Suppose a paramagnetic system that is under the action of an auxiliary (magnetic) field H. Thus, the work involved, it has to be that the work involved in a reversible process is described as đW = 10Hdm. where m is the total magnetic dipole moment of the system and u0 is the magnetic permeability of the vacuum. a) Determine the expression of the differential du of the internal energy U and indicate the two independent natural variables associated with...
5) a) Beginning with the fundamental relation, dE = Tds - PdV, show that S = S(E, D) b) Use the answer to part a to show that: c) Legendre transform S with respect to the conjugate variables (E to obtain the Helmholtz free entropy s and show that the resulting expression for dø indicates that ° = .D Note: The expression dE = Tds - PdV is sufficient proof that E = E(S.V)
02) Consider a hydrostatic system represented by the thermodynamic variables volume V, pressure P and temperature T. a) Consider entropy S = S(T, V) and derive the equation TdS. Tas = Cvat +T (1) dV. V Show that this equation can be written as follows BT TdS = CydT + PDV where Cv is the thermal capacity at constant volume, B is the isobaric expansiveness and K is isothermal compressibility: b) Consider a gas described by the equation of state...
5. (a) Derive the following Maxwell relation for a fluid system: ()-), (b) Show that for an ideal gas at a given temperature T, the chemical potential difference between an arbitrary pressure, P, and a reference pressure, Po, is given by u(P) – u(Po) = kbT In (P/Po). (c) For the case of an incompressible liquid, the volume per particle, v = V/N, is independent of the pressure; show that in this case, one has u(P) – ”(Po) = v...
2. Take the entropy to be a function of the independent variables T and V. For the van der Waals gas, derive an expression for the entropy as a function of volume at a given (fixed) temperature. Hint: You will want to take advantage of Maxwell's relations. Your answer will contain an undetermined constant of integration.
(Thermodynamics, Maxwell relations. Mathematics) 3 Consider the microcanonical formula for the equilibrium energy E(S, V,N) of some general system.58 One knows that the second deriva- tives of E are symmetric; at fixed N, we get the same answer whichever order we take partial derivatives with respect to S and V (a) Use this to show the Marwell relation (3.70) OS S,N (This should take two lines of calculus or less.) Generate two other similar formulae by taking other second partial...
1. (a) Write the total differentials of s-S(V, T),S- S(P, T). Get used to it. This is the first step in many derivations in thermodynamics. We'll actually come back to these particular ones in the next lecture or so. (b) Now, from the 1st law we know that du dq+ dw. () Take the differential along a reversible path: dU - dgrev+ dwre. [Question: why don't we write dUrev? Yes, you must answer this and the other questions in blue.]...
A mixture of two monatomic ideal gases consists of Na molecules of gas A and Na molecules of gas B in a container of volume V at temperature T. (a) Obtain an expression for natural log of the number of the accessible microstates for each species (ie, ln S, and In Ω), (b) Show that the entropy of the mixture system is 4. EA and Ea are the total energies for the two molecular species, m, and m , the...
3. In a second-order phase transition, there is no difference in entropy or volume between the two phases and therefore the Clausius-Clapeyron equation cannot be applied. Two Ehrenfest equations can, however, be found. They follow from the the equality in entropy or volume across the phase transition, and they relate dp/dT to differences in α and C, or K between the two phases (a) By setting dSdS2, find dp/dT in terms of the differences in a and Cp between the...
02) Consider a hydrostatic system represented by the thermodynamic variables volume V, pressure P and temperature T. A) A hydrostatic system undergoes a phase transition from first order to temperature To = 300 K and pressure Po = 1 x 100 Pa. In this case, it is verified that the variation of the specific volume of the system Av = 2 x 10 m3/Kg and the variation of the enthalpy specific Ah = 60 KJ/Kg. Determine the variation of the...
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