If 2 systems A and A' having initial energies Ei and Ei' respectively are thermally related and are subsequently placed in thermal contact by attaining respective final energies \(\rm E_f^0\) and \(\rm E_f^{0'}\), following a special case where initial energies lie very close to the final mean energies, which one holds true?
Q' + Q = 0
Understanding Thermal Contact and Energy Conservation
When two systems, say system A and system A', are placed in thermal contact, they can exchange energy in the form of heat until they reach thermal equilibrium. The question states that these systems are thermally related and subsequently placed in thermal contact, implying that the combined system is isolated from its surroundings for the purpose of energy exchange during the thermalization process.
Analyzing Heat Transfer in Thermal Contact
Heat transfer (Q) for a system is defined as the change in its internal energy, assuming no work is done. Let's denote the initial energy of system A as \(E_i\) and its final energy as \(E_f^0\). The heat transferred to system A is:
\(Q = E_f^0 - E_i\)
Similarly, for system A', with initial energy \(E_i'\) and final energy \(E_f^{0'}\), the heat transferred to system A' is:
\(Q' = E_f^{0'} - E_i'\)
Applying the Principle of Energy Conservation
For a combined system that is isolated, the total energy remains constant. The total energy before thermal contact is the sum of the initial energies of the two systems:
\(E_{total, initial} = E_i + E_i'\)
The total energy after thermal contact, when the systems have reached their final energies (likely at equilibrium, given the context of thermal contact), is:
\(E_{total, final} = E_f^0 + E_f^{0'}\)
According to the law of conservation of energy for an isolated system:
\(E_{total, initial} = E_{total, final}\)
Therefore,
\(E_i + E_i' = E_f^0 + E_f^{0'}\)
Deriving the Relationship Between Q and Q'
We can rearrange the energy conservation equation:
\(0 = E_f^0 - E_i + E_f^{0'} - E_i'\)
Substituting the expressions for Q and Q' that we defined earlier:
\(0 = Q + Q'\)
This gives us the relationship:
\(Q + Q' = 0\)
This equation means that the total heat exchanged between the two systems sums to zero. If one system gains heat (Q > 0), the other system must lose an equal amount of heat (Q' < 0), and vice versa. The special case mentioned, where initial energies are close to the final mean energies, doesn't change this fundamental principle of energy conservation during thermal contact in an isolated combined system.
Conclusion
The relationship that holds true when two systems A and A' are placed in thermal contact and the combined system is isolated is based on the conservation of total energy. This leads directly to the conclusion that the heat gained by one system is equal to the heat lost by the other, resulting in the sum of the heat transfers being zero.
The correct option representing this relationship is:
- \(Q' + Q = 0\)
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