Question

Suppose you had 10 identical molecules enclosed by a box. At a given instant, one molecule has an energy of 100 Joules, and the others are all stationary

(A) What is the average kinetic energy of the 10 molecules?
(B) Is this a situation of high or low entropy?
(C) What happens to the energy and entropy of the molecules in the box as time passes? Does your answer depend on the insulation of the box?

Answer 1

The average kinetic energy of the 10 molecules is 10 Joules. This is a situation of low entropy. As time passes, the energy of the molecules will spread out through collisions and interactions, increasing the entropy of the system.

Step-by-step explanation:

In this scenario, one molecule has an energy of 100 Joules and all the other molecules are stationary. To calculate the average kinetic energy, we need to consider the kinetic energies of all the molecules and then divide by the total number of molecules.

The kinetic energy of a molecule is given by the equation KE = (1/2)mv^2, where m is the mass of the molecule and v is its velocity.

Since all the molecules are identical, they have the same mass. If we assume the mass of each molecule is m, then the average kinetic energy is:

Average KE = [(1/2)(100) + (1/2)(0) + (1/2)(0) + ... + (1/2)(0)] / 10 = 10 Joules

So, the average kinetic energy of the 10 molecules is 10 Joules.

Entropy is a measure of the randomness or disorder in a system. In this scenario, since only one molecule has energy and all the other molecules are stationary, it is a situation of low entropy. There is very little randomness or disorder.

As time passes, the energy of the molecules will spread out through collisions and interactions. The molecule with 100 Joules of energy will transfer some of its energy to the stationary molecules, causing them to gain kinetic energy. The overall energy of the system will remain constant, but the distribution of energy will become more spread out. The entropy of the system will increase as the molecules become more randomly distributed in terms of their kinetic energies.

The answer does not depend on the insulation of the box. The transfer of energy and increase in entropy will occur regardless of whether the box is insulated or not.

Answer 2

A) K_average = 1/20 m v², B) low entropy , C) entropy increases.

Step-by-step explanation:

A) The kinetic energy of each molecule

K = ½ m v²

The average kinetic energy is the sum of each kinetic energy among the number of them

K_average = (½ m v² + 0 + 0 +) 10

K_average = 1/20 m v²

B) Entropy is the sum of the states of each molecule, in this configuration there are only two states one with energy and the other with zero energy, so it is a system with low entropy.

S = k ln W

Where S is the entropy, k the Bolztmann constant, W the amount of state present in the system in this case is 2

C) Let's start by analyzing the entropy, as time goes by the molecule that is moving collides with the other molecules and transfers them some energy, so the other molecules move to a different state, after a little In time all the molecules will have the same energy, each one in a different state or volume, so the number of possible state increases to 10, so the entropy increases.

Now let's analyze what happens with the energy of the system, for this case we have two possibilities

- The system is isolated, therefore as it cannot exchange energy with the environment, the total energy remains constant even when the energy of each molecule can fluctuate.

- If the system is not isolated, it can exchange energy with the environment, therefore the total energy changes, depending on the difference in energy between the molecules and the environment