If the work done on the system or by the system· is zero, which one of the following statements for a gas kept at a certain volume is correct?
Change in internal energy of the system is equal to flow of heat in or out of the system.
Understanding Thermodynamics: Work Done and Energy Change
The question asks about the relationship between the change in internal energy and the heat flow for a system where the work done is zero. This scenario is governed by the fundamental principles of thermodynamics, specifically the First Law of Thermodynamics.
The First Law of Thermodynamics Explained
The First Law of Thermodynamics is essentially a statement of the conservation of energy. It relates the change in the internal energy of a system (\(\Delta U\)) to the heat added to the system (\(Q\)) and the work done by the system (\(W\)). The standard formulation is:
\(\Delta U = Q - W\)
Where:
- \(\Delta U\) is the change in the internal energy of the system. Internal energy represents the total energy contained within the system, including kinetic and potential energy of its molecules.
- \(Q\) is the net heat transferred to the system. If heat flows into the system, \(Q\) is positive. If heat flows out, \(Q\) is negative.
- \(W\) is the net work done by the system on its surroundings. If the system does work, \(W\) is positive. If work is done on the system, \(W\) is negative.
Analyzing the Condition: Work Done is Zero
The question states that the work done on the system or by the system is zero. This means \(W = 0\). This condition often occurs in processes where the volume of the system remains constant. A thermodynamic process where the volume does not change is called an isochoric process.
Relating Internal Energy and Heat Flow When Work is Zero
Now, let's apply the condition \(W = 0\) to the First Law of Thermodynamics equation:
\(\Delta U = Q - W\)
Substitute \(W = 0\):
\(\Delta U = Q - 0\)
This simplifies the equation to:
\(\Delta U = Q\)
Conclusion on Energy Change and Heat Flow
The equation \(\Delta U = Q\) tells us that when the work done on or by the system is zero, the entire change in internal energy of the system is equal to the net heat flow into or out of the system. If heat flows in (\(Q > 0\)), the internal energy increases (\(\Delta U > 0\)). If heat flows out (\(Q < 0\)), the internal energy decreases (\(\Delta U < 0\)).
Therefore, for a gas kept at a certain volume where work done is zero, the change in internal energy of the system is equal to the flow of heat in or out of the system.
Revision Table: First Law in Different Processes
| Process Type | Description | Work Done (W) | First Law (\(\Delta U = Q - W\)) | Energy Relation |
|---|---|---|---|---|
| Isochoric | Constant volume | \(W=0\) | \(\Delta U = Q - 0\) | \(\Delta U = Q\) |
| Isobaric | Constant pressure | \(W = P\Delta V\) | \(\Delta U = Q - P\Delta V\) | \(Q = \Delta U + P\Delta V\) |
| Isothermal | Constant temperature (\(\Delta T = 0\)) | \(W\) is not zero (unless also isochoric) | \(\Delta U = 0\) for ideal gas | \(Q = W\) for ideal gas |
| Adiabatic | No heat exchange (\(Q=0\)) | \(W\) is not zero (unless also isochoric) | \(\Delta U = 0 - W\) | \(\Delta U = -W\) |
Additional Information: Internal Energy
Internal energy (\(U\)) of a system is a state function, meaning its value depends only on the current state of the system (like temperature, pressure, and volume), not on the path taken to reach that state. For an ideal gas, internal energy depends only on its temperature. Thus, for an ideal gas, if the temperature is constant (\(\Delta T = 0\)), the change in internal energy (\(\Delta U\)) is also zero.
In real gases, internal energy also has a slight dependence on pressure or volume, but temperature is the dominant factor.
Understanding internal energy is crucial for applying the First Law of Thermodynamics to analyze how energy is transferred and transformed in various thermodynamic processes involving work done and heat flow.
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