Questions and Answers
What condition must be met for work (W) to be maximum during an isothermal expansion?
Which equation represents the work done during the reversible expansion of an ideal gas?
What role does the infinitesimal step play in calculating the work done?
From which law is the expression $P_{ext} = rac{nRT}{V}$ derived?
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What indicates that the expansion of the gas is reversible?
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What happens to the internal pressure of the gas as it expands isothermally?
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What occurs when the pressure of the gas equals the external pressure during reversible expansion?
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How is the volume of a gas increased in the reversible expansion process?
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What does the equation $$W_{max} = -nRT ext{ln}(rac{V_2}{V_1})$$ represent?
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In the context of the ideal gas law, what do the variables n, R, and T stand for?
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What happens to the gas pressure during slow expansion?
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What occurs if the external pressure is lowered too much during reversible expansion?
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Study Notes
Maximum Work in Isothermal Expansion
- Consideration of n moles of an ideal gas in a cylinder with a frictionless piston.
- Isothermal and reversible expansion occurs from initial volume (V_1) to final volume (V_2) at constant temperature (T).
- External pressure (P_{ext}) is slightly less than gas pressure (P) during expansion, allowing for maximum work.
- To achieve reversibility, opposing force must be infinitesimally smaller than driving force.
- Work done is expressed as (dw = -P_{ext} dv).
Work Calculation Using Integration
- Total work done is calculated by summing infinitesimal work over the volume change, resulting in (W_{max} = \int_{V_1}^{V_2} P_{ext} dV).
- Using the ideal gas equation (PV = nRT), external pressure can be substituted as (P_{ext} = \frac{nRT}{V}).
- Integration leads to the relationship: (W_{max} = nRT \int_{V_1}^{V_2} \frac{dV}{V}).
Final Expression for Maximum Work
- Evaluating the integral gives:
- (W_{max} = nRT \ln\left(\frac{V_2}{V_1}\right)).
- This formula indicates the work done by the system during isothermal expansion.
- Alternative representation also shows the relationship:
- (W_{max} = -2.303 nRT \log\left(\frac{V_2}{V_1}\right)).
- Process occurs at constant temperature (T), emphasizing the nature of isothermal conditions.
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Description
Calculate maximum work in isothermal reversible expansion of an ideal gas from initial to final volume at constant temperature.
