A cylinder with a movable piston encloses 0.25 mole of ideal...

A cylinder with a movable piston encloses 0.25 mole of ideal gas at $300 \mathrm{~K}$. How much heat is required to increase its volume isothermally by a factor of 2 ? (a) $430 \mathrm{~J} ;$ (b) $620 \mathrm{~J}$;
(c) $900 \mathrm{~J}$;
(d) $1730 \mathrm{~J}$.

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Key Concepts

First Law of Thermodynamics

The first law of thermodynamics, stated as ?U = Q - W, establishes the principle of energy conservation in thermodynamic processes. In an isothermal process for an ideal gas, the change in internal energy (?U) is zero, leading to the conclusion that the net heat input (Q) equals the work done (W), thereby linking heat transfer directly to mechanical work.

Work Done in Isothermal Expansion

In an isothermal expansion of an ideal gas, the work done by the gas is determined by the equation W = nRT ln(V_f/V_i), where V_f and V_i are the final and initial volumes, respectively. This relation arises from integrating the pressure with respect to volume and underscores the logarithmic dependence of work on the ratio of the volumes.

Ideal Gas Law

The ideal gas law, represented as PV = nRT, relates the pressure, volume, temperature, and number of moles of an ideal gas. This equation serves as a fundamental cornerstone in thermodynamics for describing the behavior of gases where interactions between molecules are negligible.

Isothermal Process

An isothermal process is one that occurs at constant temperature. For an ideal gas undergoing such a process, the internal energy remains constant, meaning that any heat added to the system is used entirely for doing work, specifically for volume expansion or compression.

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