Statistical Interpretation of Entropy Practice Problems

A container of volume 50 cm³ is connected with a short tube fitted with a valve to a 100 cm³  container that is evacuated using a vacuum pump. The 50 cm³ container encloses 1 mole of oxygen at 290 K. Once the valve is open, the oxygen undergoes a free and isothermal expansion and fills both containers. What is the entropy change (ΔS) during this free expansion?

Two identical cylinders, A and B, are connected by a short tube with a valve. Cylinder A contains N molecules of helium, and cylinder B contains 3N molecules of argon. Initially, the valve is closed, and both gases are at the same temperature. After the valve is opened, helium and argon gases expand isothermally. Determine the average number of molecules of helium and argon that will be present in each cylinder.

Consider e = (100)^β. If a system has a total entropy S = 2.5 J/K, determine the multiplicity of the system. Express your answer in the powers of 100.

Boltzmann proposed the equation to calculate the entropy of a system in a given macrostate: S = kln(W) where W is the number of microstates corresponding to a given macrostate. Use this equation to find the entropy of each of the 4 macrostates given in the table below.

What is the probability of rolling a sum of 8 when two dice are rolled together at the same time?

If you roll two dice at the same time what will be the chance of rolling 12 as the sum of the dice?

Imagine that you need to randomly select three balls from a box that contains an equal number of red, green, and blue balls, and line them up on a table. Determine the total number of microstates. Use the table below to help you solve for your final answer.

In a bag, there is an equal amount of black, white, and grey pebbles. You have to draw out 3 pebbles at random. Construct a table showing how many microstates correspond to each macrostate. Then calculate the chance that all 3 pebbles drawn will be black.

Consider a standard deck of cards. Arrange the following five-card hands in ascending order of their probability: (a) three Aces and two Kings; (b) Four Hearts along with a Queen Spades; (c) two sevens, two eights, and a nine; and (d) any hand having at least three equal-value cards.

Suppose you are playing a game where you roll two dice at the same time. Fill the given table that displays the number of possible outcomes (microstates) for each total (macrostate). What is the likelihood of rolling a sum of 7?

Suppose you're playing a game where you roll 3 dice at once. What is the probability of rolling 3 sixes?