The equation $$q = \delta u + w$$ represents the first law of thermodynamics, which states that the heat added to a system ($$q$$) is equal to the change in internal energy ($$\delta u$$) plus the work done on the system ($$w$$). This relationship highlights the conservation of energy within a closed system, emphasizing that energy can neither be created nor destroyed but only transformed from one form to another. Understanding this equation is crucial for analyzing how energy flows in thermodynamic processes, especially regarding path dependence and independence.
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The equation illustrates that for an isolated system, if there is no heat exchange or work done, the internal energy remains constant.
The path taken during a thermodynamic process affects how much heat is exchanged and work is done, highlighting the importance of state functions versus path functions.
In an isothermal process, $$\delta u = 0$$ because the internal energy of an ideal gas depends solely on temperature, leading to $$q = w$$.
In an adiabatic process, no heat is exchanged with the surroundings, so $$q = 0$$ and therefore $$\delta u = -w$$.
The first law of thermodynamics underlines that all forms of energy are interchangeable; for example, mechanical work can be converted into thermal energy.
Review Questions
How does the first law of thermodynamics illustrate the concept of energy conservation in different thermodynamic processes?
The first law of thermodynamics encapsulated in the equation $$q = \delta u + w$$ shows that energy in a closed system is conserved. For instance, if heat is added to a system, it either increases the internal energy or does work on the surroundings. This concept illustrates that regardless of how energy changes form—whether through heat transfer or work—the total energy remains constant, reinforcing the principle of conservation across various processes.
Discuss the significance of path dependence in relation to heat and work as described by $$q = \delta u + w$$.
Path dependence refers to how the specific route taken during a thermodynamic process can influence the values of heat transfer and work done. For example, in processes such as expansion or compression, different paths (isothermal vs. adiabatic) yield different amounts of heat and work despite starting and ending at the same states. This emphasizes that while internal energy change ($$\delta u$$) is a state function and independent of the path taken, both heat ($$q$$) and work ($$w$$) are path-dependent variables.
Evaluate how understanding the relationship expressed by $$q = \delta u + w$$ helps in practical applications like engines or refrigerators.
Understanding this relationship allows for better design and optimization of engines and refrigerators by analyzing how efficiently they convert heat into work or vice versa. For instance, in an engine cycle, knowing how much work is produced from heat absorbed can lead to improvements in fuel efficiency and performance. Similarly, in refrigeration cycles, it helps engineers minimize energy consumption while maximizing heat removal. By applying the principles from $$q = \delta u + w$$, we can achieve more effective thermal management in various technological applications.