A 2 g bullet flying at 390 m/s hits a 2.7 kg aluminum block at room temperature and embeds itself in the aluminum. After everything has come to equilibrium, how much has the entropy of the aluminum block increased?
To determine the increase in entropy of the aluminum block, we need to consider the change in entropy caused by the embedding of the bullet. Entropy is a measure of the randomness or disorder in a system.
First, let's calculate the initial entropy of the aluminum block. Entropy can be calculated using the equation:
ΔS = m * Cp * ln(Tf/Ti)
Where ΔS is the change in entropy, m is the mass of the system, Cp is the specific heat capacity, ln represents natural logarithm, Tf is the final temperature, and Ti is the initial temperature.
In this case, the bullet comes to rest and equilibrates its thermal energy with the aluminum block at room temperature. So, the initial temperature of the aluminum block is the same as the final temperature, which we can consider as the room temperature.
The specific heat capacity of aluminum is approximately 0.897 J/g·°C, and the mass of the aluminum block is 2.7 kg. Assuming room temperature is around 25°C, we have:
ΔS = (2.7 kg) * (0.897 J/g·°C) * ln(25/25)
= 0
The initial entropy of the aluminum block remains the same, as there is no change in its initial state.
Now, let's consider the change in entropy caused by the embedding of the bullet. Since the bullet embeds itself into the aluminum block, it increases the randomness or disorder of the system. This change in entropy can be calculated using the equation:
ΔS = m_bullet * ln(V/V_0)
Where ΔS is the change in entropy, m_bullet is the mass of the bullet, V is the final volume after embedding, and V_0 is the initial volume of the bullet.
Given that the mass of the bullet is 2 g, and assuming it is a point mass, we can consider its initial volume as negligible.
ΔS = (2 g) * ln(V/0)
= (2 g) * ln(∞)
= (∞)
As the volume of the embedded bullet approaches infinity, the change in entropy also approaches infinity, meaning the embedding of the bullet significantly increases the randomness or disorder of the system.
Therefore, the entropy of the aluminum block has increased by an infinite amount after the bullet embeds itself in the aluminum.