In the equation A and B are constants, the other variables have their usual meanings. (a) P V N k B T A2 N 2 N (b) P V 2 N k B T V B 2 2 (c) P m V N2 (V 2N B) NA!k B T 2 2 (d) P AVN2 (V 2N B) k B T N2 (e) P V (V 2N B) N k B 2 (f) P N V N N Bk B T V2 2 1 N (g) P N N k B T 2 V V 2 1 (h) 2B P N 3N k B T 2 V V N2 (i) P 2 (V 2N B) N k B T V Answer: Since we want the equation of state, calculate the pressure in terms of volume and ln Q temperature using P k B T N, T .
Determine the average energy, h Ei, for a monatomic ideal gas given the partition function for this gas, 1 Q(N, V, T ) N! N V h Ei k B T (c) h Ei k B T 2 (d) h Ei k B T N V (e) h Ei k B T 3N 2 (f) h Ei h2 N V 3N 2T Answer: First, rearrange this equation, 1 Q(N, V, T ) T N! Take the natural log of both sides of the partition function and separate the terms to obtain 3N 3N ln T ln N ln V ln Q(N, V, T ) N !
2 2 h2 Note that only the second term in the above equation depends on T .
University of Minnesota Statistical Molecular Thermodynamics Homework Week 3 1.
You are given the following partition function for a fictitious gas What is the correct equation of state for Vikonium?
It will be helpful to make use of the geometric series, X xj 1 to derive this result.
e (a) Q (b) Q (c) Q (d) Q (e) Q (f) Q 1 Answer: The partition function for a harmonic oscillator is given as qho (T ) X e 21 X If we let x we can simplify the partition function to qho (T ) X Note that this summation is the geometric series, X xj X 1 . The average energy for a monatomic van der Waals gas is 3 a NA2 h Ei NA k B T 2 V Use the definition of the constant volume molar heat capacity discussed in lecture video 3.4 to determine a formula for the constant volume molar heat capacity of a monatomic van der Waals gas.Start separating ln Q into pieces that include V and those that do not, ln Q ln no V N ln (V 2N B) P k B T ln Q P N 2 k B T 0 2 V 2N B V N2 P 2 (V 2N B) N k B T V N, T 2 N k B T N 2, V 2N B V 2 V 2.In an Einstein crystal, one principal assumption is that each of the N atoms of the crystal vibrate independently about their lattice positions.We can proudly state that our thermodynamics homework help online works with experts who obtain a higher degree in many fields.When people hire for the jobs we propose, they go through multiple tests and interviews to prove they are really the best.At a fixed temperature this means that the partition function will increase.Thermodynamics is a branch of physics which studies the laws of govern the conversion of energy from one form to another.Substituting into the equation for the probability 1 2 ) pj qvib qvib qvib We have shown that the partition function for the harmonic oscillator is qvib 1 Insert this into the equation for the probability pj (1 ) (1 ) From the preceding equation we can see that for the ground state p0 (1 ) 1 Therefore, the fraction in the ground state is equal to this probability.f0 p0 1 1 T To find the fraction of harmonic oscillators in the ground state at different temperatures, first convert the fundamental vibrational frequency for N2 to . c (3 1010 cm )(2330 ) 6.99 1013 We can then obtain a value for : (6.626 J s)(6.99 1013 ) 3352 K k B 1.38 J This simplifies the equation for the fraction of harmonic oscillators to: f0 1 T 1 Insert the different temperatures for T to obtain the fraction in the ground state: Temperature (K) f0 300 0.9650 10.It gets harder when you have a lot of technical tasks in classes like physics, mechanics, or thermodynamics.Well, there is a solution: our thermodynamics homework help.