Question

Entropy is defined in terms of microstates. We often illustrate how this works using a small number of discrete objects like coins or dice. When rolling three dice, 1+2+6 and 1+6+2 and 3+2+4 are different microstates, even though each adds up to 9. There is only one microstate that adds to 3, but there are 25 that add to 9. That is why you are much more likely to roll a 9 than a 3.

With coin-like objects that have two choices, counting the number of possible microstates is illustrated by Pascal's triangle. The branch of mathematics that deals with counting such things is called combinatorics. In any system of more than a handful of atoms the number of microstates becomes larger than the number of protons in the universe. So we typically calculate changes in entropy, with an eye toward whether it is positive or negative.
But in this problem we'll count microstates.
You flip 6 coins.
a.How many microstates are there?
__________
b.What is the number of heads that has the highest number of microstates? If two or more have the same number, give the one with the lowest number of heads
_____________.
c.How many microstates are there that have exactly one head?
_____________
d.How many more likely is it that you get the most likely number of heads than that you get one head?
_____________
A block made of nickel with a mass of 0.90 kg is heated to 850°C, then dropped into 5.00 kg of water at 11°C. What is the total change in entropy (in J/K) of the block-water system, assuming no energy is lost by heat from this system to the surroundings? The specific heat of nickel is 440 J/(kg K), and the specific heat of water is 4,186 J/(kg - K). (Hint: note that dQ = med.) ____________3/K

Answer 1

Six flipped coins have 64 microstates. The highest number of microstates occurs for 3 heads or 3 tails due to symmetry, and there are 6 microstates for exactly one head. Calculating the change in entropy for the nickel-water system requires the final equilibrium temperature, which is not provided.

Step-by-step explanation:

For a system of 6 coins being flipped, we calculate the number of microstates as follows:

The number of heads with the highest number of microstates can be determined by finding the middle terms in the binomial expansion, which corresponds to either 3 heads or 3 tails, each having the same highest number of microstates. Because heads and tails are symmetrical, the middle terms represent the combinations for 3 heads or 3 tails in this case.

There are C(6,1) = 6 ways to get exactly one head since there are 6 coins, and we can choose any one of them to be heads.

Determining how many more likely it is to get the most likely number of heads rather than one head involves calculating the ratio of the corresponding microstates:

The change in entropy for the nickel and water system isn't possible to calculate without knowing the final temperature after thermal equilibrium is reached. The formula dQ = mcdT could be used to find the change in heat, and subsequently, the entropy change could be determined using ΔS = ∑(dQ/T), where ΔS is the entropy change, dQ is the heat change, m is the mass, c is the specific heat, and dT is the change in temperature. Without the final equilibrium temperature, the entropy change cannot be calculated from the given information.