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1. Determine the Helmholtz free energy (Tdependent) of a two state system. Let one state have...
3. [10 marks] We've covered the definition of Gibbs free energy, Helmholtz free energy, entropy, and...
3. [10 marks] We've covered the definition of Gibbs free energy, Helmholtz free energy, entropy, and enthalpy. There is also something called their fundamental equations. a) [8 marks] Use online resources and your peers to determine the total differential form of each of these terms. For example, if we're looking at Gibbs free energy, defined as G = H – TS, then determine what dG would be. b) [2 marks] Why is it useful to use this form? c) [+2...
Finding conditions for Helmholtz free energy 5.8. The steps in the strategy for finding conditions for...
Finding conditions for Helmholtz free energy
5.8. The steps in the strategy for finding conditions for equilibrium are: a) Write an expression for the change in entropy of the system when it is taken through an arbitrary process. (b) Write the isolation constraints in differential form. (c) Use the isolation constraints to eliminate dependent variables in the expression for the entropy. (a) Collect terms. (e) Set the coefficients of each differential equal to zero. () Solve these equations for the...
2. (25 pts) Derive the (a) Maxwell relation for the Helmholtz Free Energy F=U-TS. Show ALL...
2. (25 pts) Derive the (a) Maxwell relation for the Helmholtz Free Energy F=U-TS. Show ALL steps and justifications in your derivation. Using your result in (a) comment on how (b) the entropy behaves for an isothermal expansion of an ideal gas. Finally, show the validity of the following equations (c) U = F-TOOF) -T2 and at (T) 01 (d) C =-1(
1. Show that for a classical ideal gas, Q1 alnQ1 NK Hint: Start with the partition function for t...
1. Show that for a classical ideal gas, Q1 alnQ1 NK Hint: Start with the partition function for the classical ideal gas ( Q1) and use above equation to find the value of right-hand side and compare with the value of r we derive in the class. (Recall entropy you derived for classical gas) NK Making use of the fact that the Helmholtz free energy A (N, V, T) of a thermodynamic system is an extensive property of the system....
3. Derive the following relationship between the Helmholtz free energy F and the partition function Z for a system of N particles: (a) Starting with the thermodynamic definition F-U-TS, substitute th...
3. Derive the following relationship between the Helmholtz free energy F and the partition function Z for a system of N particles: (a) Starting with the thermodynamic definition F-U-TS, substitute the statistical mechanics results which give U and S in terms of occupation numbers n, state energies e and the most probable number of microstates t* to find, (b) Write out texplicitly in terms of occupation numbers using Stirling's approxima- tion (check the Lagrange multiplier derivation of the Boltzmann distribution)...
. (40 points) Consider an insulated container of volume V2. N idea gas molecules are initially...
. (40 points) Consider an insulated container of volume V2. N idea gas molecules are initially confined within volume V, by a piston and the remaining volume V2 - Vi is in vacuum. Let T,, P1, E1, S, Al, Hi, G, be the temperature, pressure, energy, entropy, Helmholtz free energy, enthalpy, and Gibbs free energy of the ideal gas at this state, respectively V1 V2-V Using Sackur-Tetrode equation for the entropy of ideal gas where kB R/NA is Boltzmann's constant...
Problem 8.3 - A New Two-State System Consider a new two-level system with a Hamiltonian given...
Problem 8.3 - A New Two-State System Consider a new two-level system with a Hamiltonian given by i = Ti 1461 – 12) (2) (3) Also consider an observable represented by the operator Ŝ = * 11/21 - *12/11: It should (hopefully) be clear that 1) and 2) are eigenkets of the Hamiltonian. Let $1) be an eigenket of S corresponding to the smaller eigenvalue of S and let S2) be an eigenket of S corresponding to the larger eigenvalue....
Please explain your answer Consider a two-level system where the ground state has an energy of...
Please explain your answer
Consider a two-level system where the ground state has an energy of O kJ mol-and is non- degenerate, and the higher state has an energy of ε kJ mol- and is triply degenerate. What is the population of the ground state at temperature T (Kelvin)? Select one: o a. 3 / (1 + exp(-3ɛ/kT)) o b.3/(1 + 3exp(-ɛ/kT)) c. 1/(1 + 3exp(-ɛ/kT)) O d. 1/(3 + exp(-ɛ/KT))
Consider a two-level system where the ground state has an energy of 0 kJ mol-1 and is non-degenerate, and the higher st...
Consider a two-level system where the ground state has an energy of 0 kJ mol-1 and is non-degenerate, and the higher state has an energy of ε kJ mol-1 and is triply degenerate. What is the population of the ground state at temperature I (Kelvin)? Select one: o a. 17(3 + exp(-ɛ/kT)) b. 1/(1 + 3exp(-€/kT)) O c. 3/(1 + 3exp(-ɛ/kT)) O O d. 3/(1 + exp(-3ɛ/kT))
Thermodynamics Consider an insulated container of volume V2. N ideal gas molecules are initially confined within...
Thermodynamics
Consider an insulated container of volume V2. N ideal gas molecules are initially confined within a sub-volume (V1) by a piston and the remaining volume V2 - Viis in vacuum. Let T., P., U1, S1, A1, H1, and G1 be the temperature, pressure, internal energy, entropy, Helmholtz free energy, enthalpy, and Gibbs free energy of the ideal gas at this state, respectively. Now, imagine that the piston is removed so that the gas has volume V2. After some time...
3. [10 marks] We've covered the definition of Gibbs free energy, Helmholtz free energy, entropy, and enthalpy. There is also something called their fundamental equations. a) [8 marks] Use online resources and your peers to determine the total differential form of each of these terms. For example, if we're looking at Gibbs free energy, defined as G = H – TS, then determine what dG would be. b) [2 marks] Why is it useful to use this form? c) [+2...
Finding conditions for Helmholtz free energy
5.8. The steps in the strategy for finding conditions for equilibrium are: a) Write an expression for the change in entropy of the system when it is taken through an arbitrary process. (b) Write the isolation constraints in differential form. (c) Use the isolation constraints to eliminate dependent variables in the expression for the entropy. (a) Collect terms. (e) Set the coefficients of each differential equal to zero. () Solve these equations for the...
2. (25 pts) Derive the (a) Maxwell relation for the Helmholtz Free Energy F=U-TS. Show ALL steps and justifications in your derivation. Using your result in (a) comment on how (b) the entropy behaves for an isothermal expansion of an ideal gas. Finally, show the validity of the following equations (c) U = F-TOOF) -T2 and at (T) 01 (d) C =-1(
1. Show that for a classical ideal gas, Q1 alnQ1 NK Hint: Start with the partition function for the classical ideal gas ( Q1) and use above equation to find the value of right-hand side and compare with the value of r we derive in the class. (Recall entropy you derived for classical gas) NK Making use of the fact that the Helmholtz free energy A (N, V, T) of a thermodynamic system is an extensive property of the system....
3. Derive the following relationship between the Helmholtz free energy F and the partition function Z for a system of N particles: (a) Starting with the thermodynamic definition F-U-TS, substitute the statistical mechanics results which give U and S in terms of occupation numbers n, state energies e and the most probable number of microstates t* to find, (b) Write out texplicitly in terms of occupation numbers using Stirling's approxima- tion (check the Lagrange multiplier derivation of the Boltzmann distribution)...
. (40 points) Consider an insulated container of volume V2. N idea gas molecules are initially confined within volume V, by a piston and the remaining volume V2 - Vi is in vacuum. Let T,, P1, E1, S, Al, Hi, G, be the temperature, pressure, energy, entropy, Helmholtz free energy, enthalpy, and Gibbs free energy of the ideal gas at this state, respectively V1 V2-V Using Sackur-Tetrode equation for the entropy of ideal gas where kB R/NA is Boltzmann's constant...
Problem 8.3 - A New Two-State System Consider a new two-level system with a Hamiltonian given by i = Ti 1461 – 12) (2) (3) Also consider an observable represented by the operator Ŝ = * 11/21 - *12/11: It should (hopefully) be clear that 1) and 2) are eigenkets of the Hamiltonian. Let $1) be an eigenket of S corresponding to the smaller eigenvalue of S and let S2) be an eigenket of S corresponding to the larger eigenvalue....
Please explain your answer
Consider a two-level system where the ground state has an energy of O kJ mol-and is non- degenerate, and the higher state has an energy of ε kJ mol- and is triply degenerate. What is the population of the ground state at temperature T (Kelvin)? Select one: o a. 3 / (1 + exp(-3ɛ/kT)) o b.3/(1 + 3exp(-ɛ/kT)) c. 1/(1 + 3exp(-ɛ/kT)) O d. 1/(3 + exp(-ɛ/KT))
Consider a two-level system where the ground state has an energy of 0 kJ mol-1 and is non-degenerate, and the higher state has an energy of ε kJ mol-1 and is triply degenerate. What is the population of the ground state at temperature I (Kelvin)? Select one: o a. 17(3 + exp(-ɛ/kT)) b. 1/(1 + 3exp(-€/kT)) O c. 3/(1 + 3exp(-ɛ/kT)) O O d. 3/(1 + exp(-3ɛ/kT))
Thermodynamics
Consider an insulated container of volume V2. N ideal gas molecules are initially confined within a sub-volume (V1) by a piston and the remaining volume V2 - Viis in vacuum. Let T., P., U1, S1, A1, H1, and G1 be the temperature, pressure, internal energy, entropy, Helmholtz free energy, enthalpy, and Gibbs free energy of the ideal gas at this state, respectively. Now, imagine that the piston is removed so that the gas has volume V2. After some time...
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