Consider a two level system (such as the orientation of the magnetic moments of protons in a magnetic field, e.g. NMR). Let the spacing between the two levels be delta that is, the lower level has an energy of 0 and the other an energy of delta. Evaluate the following quantities in terms of delta and T.
a) the partition function
b) the internal energy
c) the entropy
In the limit of large T, does the system in the previous problem conform with the equipartition principle? Should it?
The partition function for a general system is:
Z = Sum over all states of Exp(-beta Energy of the state)
where beta = 1/(kT)
The probability of the system to be in some particular state is:
P(state) = Exp[-beta E(state)]/Z
In this case we have two states, one with energy 0 and another with energy delta. This means that:
Z = Exp(0) + Exp(-beta delta) = 1 + exp(-beta delta) (1)
The probability for the system to be in the state with energy 0 is:
P(0) = Exp(0)/[1 + exp(-beta delta)]
The probability for the ...
A detailed solution is derived from first principles.
Cell Biology and its Internal Components
1. Identify two different types of organisms that you have seen interacting, such as bees and flowers. Now form a simple hypothesis about this interaction. Use the scientific method and your imagination to design an experiment that tests this hypothesis. Be sure to identify variables and a control for them.
Select a molecule. List the atoms that that molecule is composed of and describe the type of bond that holds those atoms together. Be sure to explain how this bond works.
1. The integrity of the plasma membrane is essential for cellular survival. Could the immune system utilize this fact to destroy foreign cells that have invaded the body? How might cells of the immune system disrupt membranes of foreign cells? (Two hints: virtually all cells can secrete proteins, and some proteins form pores in membranes.)
2. Most cells are very small. What physical and metabolic constraints limit cell size? What problems would an enormous cell encounter? What adaptations might help a very large cell to survive?
3. When a brown bear eats a salmon, does the bear acquire all the energy contained in the body of the fish? Why or why not? What implications do you think this answer would have for the relative abundance (by weight) of predators and their prey? Does the second law of thermodynamics help explain the title of the book, Why Big Fierce Animals are Rare?
4. You are called before the Ways and Means Committee of the House of Representatives to explain why the U.S. Department of Agriculture should continue to fund photosynthesis research. How would you justify the expense of producing, by genetic engineering, the enzyme that catalyzes the reaction of RuBP with CO2 and prevents RuBP from reacting with oxygen as well as CO2? What are the potential applied benefits of this research?
Some species of bacteria that live at the surface of sediment on the bottom of lakes are facultative anaerobes; that is, they are capable of either aerobic or anaerobic respiration. How will their metabolism change during the summer when the deep water becomes anoxic (deoxygenated)? If the bacteria continue to grow at the same rate, will glycolysis increase, decrease, or remain the same after the lake becomes anoxic? Explain why.