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06- Using micro canonical ensemble, a) Verify the zeroth and first laws of thermodynamics. b) Show...
Pb2. Consider the case of a canonical ensemble of N gas particles confined to a t...
Pb2. Consider the case of a canonical ensemble of N gas particles confined to a t rectangular parallelepiped of lengths: a, b, and c. The energy, which is the translational kinetic energy, is given by: o a where h is the Planck's constant, m the mass of the particle, and nx, ny ,nz are integer numbers running from 1 to +oo, (a) Calculate the canonical partition function, qi, for one particle by considering an integral approach for the calculation of...
Need help on this thermodynamics question. Thanks Data given from function Cp=22.64 + 6.28 x (10^-3)*T...
Need help on this thermodynamics question. Thanks
Data given from function Cp=22.64 + 6.28 x (10^-3)*T [Jmol-1 K-1
]
Cp(J/mol.K) T(K) 24.524 300 24.838 350 400 25.152 25.466 450 500 25.78 550 26.408 26.722 650 27.036 700 750 27.35 27.664 800 27.978 28.292 28.606 950 1000 28.92 1050 29.234 1100 29.548 1150 29.862 30.176 1200 1250 30.49 1300 30.804 1350 31.118 358 31.16824 The specific heat capacity of solid copper above 300 K is given by Cp-22.64+6.28 x 103TJmol K1]...
Question 10 Statistical thermodynamics may be used to find the radiation pressure P for cavity (or...
Question 10 Statistical thermodynamics may be used to find the radiation pressure P for cavity (or black body) radiation in terms of the energy per unit volume u. (a) An ideal quantum gas comprises non-interacting identical particles with discrete quantum states labelled 1, 2, ...,r ,....The partition function is given by Z (T,V,N)- > exp(-B(n,&, + п,&, +...)} пп. (i) Define the symbols n1, n2,...,n,...and 81, 82, ..., Er,... (iiExplain why, for photons, the partition function may be expressed as:...
1. Quick Exercises (a) In lecture 15, we showed that the canonical partition function, Q, is...
1. Quick Exercises (a) In lecture 15, we showed that the canonical partition function, Q, is related to the Helmholz free energy: A = -kTinQ. Using the fundamental thermo- dynamic relation of Helmheltz free energy i.e. dA = -SAT - PDV + pdN), express P, and p in terms of Q. (b) The canonical partition function for N non-interacting, indistinguishable parti- cles in volume V at temperature T is given by Q(N,V,T) = where where q(V.T) is the partition function...
1. Quick Exercises (a) In lecture 15, we showed that the canonical partition function, Q, is related to the Helmhol...
1. Quick Exercises (a) In lecture 15, we showed that the canonical partition function, Q, is related to the Helmholz free energy: A = -kTinQ. Using the fundamental thermo- dynamic relation of Helmheltz free energy (i.e. dA = -SIT - PdV + pdN), express P, and u in terms of Q. (b) The canonical partition function for N non-interacting, indistinguishable parti- cles in volume V at temperature T is given by Q(N, V,T) = where where 9(V, T) is the...
1. Quick Exercises (a) In lecture 15, we showed that the canonical partition function, Q, is related to the Helmholz fr...
1. Quick Exercises (a) In lecture 15, we showed that the canonical partition function, Q, is related to the Helmholz free energy: A = -kTinQ. Using the fundamental thermo- dynamic relation of Helmheltz free energy i.e. dA = -SAT - PDV + udN), express P, and u in terms of Q. (b) The canonical partition function for N non-interacting, indistinguishable parti- cles in volume V at temperature T is given by ON Q(N,V,T) = where where q(VT) is the partition...
Return to the original model. We now introduce a Poisson intensity parameter X for every time point and denote the parameter () that gives the canonical exponential family representation as above by...
Return to the original model. We now introduce a Poisson intensity parameter X for every time point and denote the parameter () that gives the canonical exponential family representation as above by θ, . We choose to employ a linear model connecting the time points t with the canonical parameter of the Poisson distribution above, i.e., n other words, we choose a generalized linear model with Poisson distribution and its canonical link function. That also means that conditioned on t,...
10. a) Show that, for a system in equilibrium with constant internal energy, the first and...
10. a) Show that, for a system in equilibrium with constant internal energy, the first and second laws together require that dS P dVT b) Determine for an ideal gas. What are the dimensions of the answer? c) Does the answer to part b make sense given that equilibrium is also the state with the maximum value for the microscopic entropy, defined as S = kg logo?
10. a) Show that, for a system in equilibrium with constant internal energy, the first and...
10. a) Show that, for a system in equilibrium with constant internal energy, the first and second laws together require that dS P dv = 1 b) Determine as for an ideal gas. What are the dimensions of the answer? c) Does the answer to part b make sense given that equilibrium is also the state with the maximum value for the microscopic entropy, defined as S = kp log w?
4. Show all your work in this exercise - copying the results from the list with...
4. Show all your work in this exercise - copying the results from the list with canonical distributions is not sufficient. (a) Consider the Gamma(α, β) distribution (α, β > 0), The MGF for Y ~ Gamma(α, β) is given by my(t) compute EY] and VarlY] (1-St)-α . Use this MGF to (b) The geometric distribution Geom(p) is defined by (1-pf-ıp, p(k) = P(X = k) k=1,2, Show that the MGF for X~Geom(p) is given by pe mx(t)-1-(1 -p)et (c)...
Pb2. Consider the case of a canonical ensemble of N gas particles confined to a t rectangular parallelepiped of lengths: a, b, and c. The energy, which is the translational kinetic energy, is given by: o a where h is the Planck's constant, m the mass of the particle, and nx, ny ,nz are integer numbers running from 1 to +oo, (a) Calculate the canonical partition function, qi, for one particle by considering an integral approach for the calculation of...
Need help on this thermodynamics question. Thanks
Data given from function Cp=22.64 + 6.28 x (10^-3)*T [Jmol-1 K-1
]
Cp(J/mol.K) T(K) 24.524 300 24.838 350 400 25.152 25.466 450 500 25.78 550 26.408 26.722 650 27.036 700 750 27.35 27.664 800 27.978 28.292 28.606 950 1000 28.92 1050 29.234 1100 29.548 1150 29.862 30.176 1200 1250 30.49 1300 30.804 1350 31.118 358 31.16824 The specific heat capacity of solid copper above 300 K is given by Cp-22.64+6.28 x 103TJmol K1]...
Question 10 Statistical thermodynamics may be used to find the radiation pressure P for cavity (or black body) radiation in terms of the energy per unit volume u. (a) An ideal quantum gas comprises non-interacting identical particles with discrete quantum states labelled 1, 2, ...,r ,....The partition function is given by Z (T,V,N)- > exp(-B(n,&, + п,&, +...)} пп. (i) Define the symbols n1, n2,...,n,...and 81, 82, ..., Er,... (iiExplain why, for photons, the partition function may be expressed as:...
1. Quick Exercises (a) In lecture 15, we showed that the canonical partition function, Q, is related to the Helmholz free energy: A = -kTinQ. Using the fundamental thermo- dynamic relation of Helmheltz free energy i.e. dA = -SAT - PDV + pdN), express P, and p in terms of Q. (b) The canonical partition function for N non-interacting, indistinguishable parti- cles in volume V at temperature T is given by Q(N,V,T) = where where q(V.T) is the partition function...
1. Quick Exercises (a) In lecture 15, we showed that the canonical partition function, Q, is related to the Helmholz free energy: A = -kTinQ. Using the fundamental thermo- dynamic relation of Helmheltz free energy (i.e. dA = -SIT - PdV + pdN), express P, and u in terms of Q. (b) The canonical partition function for N non-interacting, indistinguishable parti- cles in volume V at temperature T is given by Q(N, V,T) = where where 9(V, T) is the...
1. Quick Exercises (a) In lecture 15, we showed that the canonical partition function, Q, is related to the Helmholz free energy: A = -kTinQ. Using the fundamental thermo- dynamic relation of Helmheltz free energy i.e. dA = -SAT - PDV + udN), express P, and u in terms of Q. (b) The canonical partition function for N non-interacting, indistinguishable parti- cles in volume V at temperature T is given by ON Q(N,V,T) = where where q(VT) is the partition...
Return to the original model. We now introduce a Poisson intensity parameter X for every time point and denote the parameter () that gives the canonical exponential family representation as above by θ, . We choose to employ a linear model connecting the time points t with the canonical parameter of the Poisson distribution above, i.e., n other words, we choose a generalized linear model with Poisson distribution and its canonical link function. That also means that conditioned on t,...
10. a) Show that, for a system in equilibrium with constant internal energy, the first and second laws together require that dS P dVT b) Determine for an ideal gas. What are the dimensions of the answer? c) Does the answer to part b make sense given that equilibrium is also the state with the maximum value for the microscopic entropy, defined as S = kg logo?
10. a) Show that, for a system in equilibrium with constant internal energy, the first and second laws together require that dS P dv = 1 b) Determine as for an ideal gas. What are the dimensions of the answer? c) Does the answer to part b make sense given that equilibrium is also the state with the maximum value for the microscopic entropy, defined as S = kp log w?
4. Show all your work in this exercise - copying the results from the list with canonical distributions is not sufficient. (a) Consider the Gamma(α, β) distribution (α, β > 0), The MGF for Y ~ Gamma(α, β) is given by my(t) compute EY] and VarlY] (1-St)-α . Use this MGF to (b) The geometric distribution Geom(p) is defined by (1-pf-ıp, p(k) = P(X = k) k=1,2, Show that the MGF for X~Geom(p) is given by pe mx(t)-1-(1 -p)et (c)...
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