Chapter 2.6, Problem 28P

How many possible arrangements are there for a deck of 52 playing cards? (For simplicity, consider only the order of the cards, not whether they are turned upside-down, etc.) Suppose you start with a sorted deck and shuffle it repeatedly, so that all arrangements become “accessible.” How much entropy do you create in the process? Express your answer both as a pure number (neglecting the factor of k) and in SI units. Is this entropy significant compared to the entropy associated with arranging thermal energy among the molecules in the cards?

To determine

To Find: Possible arrangements of the deck of 52 cards. The entropy while shuffling the card. Significance of the entropy of the card and the thermal energy of the molecule of the card.

Answer to Problem 28P

  $52!=8.06\times {10}^{67},2.16\times {10}^{-21}J/K$

Explanation of Solution

Given:

A deck of 52 cards.

Formula Used:

  $S=Nk\ln \Omega$

Calculation:

The possibility of a card to be on 1^st position $=52$

The possibility of a card to be on 2^nd position $=51$

The possibility of a card to be on 3^rd position $=50$

Similarly

The possibility of a card to be on 52^ndposition $=1$

The total number of possible ways of arranging the card is

  $52!=8.06\times {10}^{67}$

As all the arrangements are accessible.

Entropy while shuffling the card is

  $\begin{array}{l}\frac{S}{k}=\ln 52!=156;\\ S=156k=2.16\times {10}^{-21}J/K\end{array}$

As the entropy is very negligible if compared to the entropy of particles due to thermal motion of the card while shuffling the cards.

Conclusion:

Thus, possibility of arrangements of cards and entropy during that are $52!=8.06\times {10}^{67},2.16\times {10}^{-21}J/K$ respectively.