A bit of computer memory is some physical object that can be in two different states, often interpreted as 0 and 1. A byte is eight bits, a kilobyte is bytes, a megabyte is 1024 kilobytes, and a gigabyte is 1024 megabytes.
(a) Suppose that your computer erases or overwrites one gigabyte of memory, keeping no record of the information that was stored. Explain why this process must create a certain minimum amount of entropy, and calculate how much.
(b) If this entropy is dumped into an environment at room temperature, how much heat must come along with it? Is this amount of heat significant?
(a) The entropy created is .
(b) The amount of heat generated is .
Computer memory can be classified in two different states, often defined as 0 and 1 .
The expression for entropy with multiplicity is given as:
= Boltzmann constant
= the number of atoms
If a single bit of computer memory is taken to be a particle with two states, then a collection of bits has a particle , and its entropy may be determined using equation (1)
If a gigabyte is used to store certain information and later erased without a backup, of information is lost indirectly. If the original information is replaced with a known pattern, it appears that the entropy hasn't changed because the state has only changed. If the original information containing the random pattern is deleted, entropy is likely to increase.
The amount of entropy generated by randomizing gigabytes can be given as:
By substituting the value of in the above equation, we get,
Hence, the required entropy is .
The amount of heat is given as:
By substituting the values in the above equation, we get,
Hence, the amount of heat generated is .
In Problem you computed the entropy of an ideal monatomic gas that lives in a two-dimensional universe. Take partial derivatives with respect to , and N to determine the temperature, pressure, and chemical potential of this gas. (In two dimensions, pressure is defined as force per unit length.) Simplify your results as much as possible, and explain whether they make sense.
Suppose you have a mixture of gases (such as air, a mixture of nitrogen and oxygen). The mole fraction of any species is defined as the fraction of all the molecules that belong to that species: . The partial pressure of species is then defined as the corresponding fraction of the total pressure: . Assuming that the mixture of gases is ideal, argue that the chemical potential of species in this system is the same as if the other gases were not present, at a fixed partial pressure .
A cylinder contains one liter of air at room temperature ( ) and atmospheric pressure . At one end of the cylinder is a massless piston, whose surface area is . Suppose that you push the piston in very suddenly, exerting a force of . The piston moves only one millimeter, before it is stopped by an immovable barrier of some sort.
(a) How much work have you done on this system?
(b) How much heat has been added to the gas?
(c) Assuming that all the energy added goes into the gas (not the piston or cylinder walls), by how much does the internal energy of the gas increase?
(d) Use the thermodynamic identity to calculate the change in the entropy of the gas (once it has again reached equilibrium).
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