Chapter 19: Problem 58
In a cylinder, 1.20 mol of an ideal monatomic gas, initially at 3.60 \(\times\) 10\(^5\) Pa and 300 K, expands until its volume triples. Compute the work done by the gas if the expansion is (a) isothermal; (b) adiabatic; (c) isobaric. (d) Show each process in a \(pV\)-diagram. In which case is the absolute value of the work done by the gas greatest? Least? (e) In which case is the absolute value of the heat transfer greatest? Least? (f) In which case is the absolute value of the change in internal energy of the gas greatest? Least?
Short Answer
Step by step solution
Identify Given Information
Step 2(a): Calculate Work for Isothermal Expansion
Step 2(b): Calculate Work for Adiabatic Expansion
Step 2(c): Calculate Work for Isobaric Expansion
Compare Work Done
Compare Heat Transfer and Change in Internal Energy
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Isothermal Expansion
- \( W = nRT \ln\left( \frac{V_2}{V_1} \right) \)
This makes isothermal processes unique as it means the system exchanges the most heat with its surroundings among all types of thermodynamic processes.
Adiabatic Expansion
- \( PV^\gamma = \text{constant} \)
Adiabatic processes can be fast because there's no concern about the heat exchange.
This process shows the greatest change in temperature and internal energy, balancing the reduced work done compared to isothermal expansion.
Isobaric Process
- \( W = P \Delta V \)
Though the amount of work done by the gas is more than in an adiabatic expansion, it is still less than the work done during an isothermal expansion when all other factors remain constant.
Work Done by Gas
- Isothermal: Maximum work done as pressure decreases slowly over a large distance because temperature remains constant.
- Adiabatic: Least work done because of no heat exchange and subsequent rapid volume change.
- Isobaric: Intermediate work done, as pressure stays the same throughout the volume change.
Heat Transfer Calculations
- Isothermal: Heat transfer is non-zero and equals the work done, making it substantial. This is because temperature remains constant and heat compensates for the work done by the gas.
- Adiabatic: There is no heat transfer. All energy changes occur internally through work done by or on the gas.
- Isobaric: The heat transfer depends on both work done and changes in internal energy. Due to constant pressure, heat must be added or removed to change temperature.