One mole of an ideal gas undergoes an isothermal expansion....

One mole of an ideal gas undergoes an isothermal expansion. Find the thermal energy $Q$ added to the gas in terms of the initial and final volumes and the temperature. (Hint:
Use the first law of thermodynamics.)

Key Concepts

Heat Transfer in Isothermal Processes

In an isothermal process where the internal energy remains constant (?U = 0), the heat added to or removed from the system must equal the work done by or on the system. This means that for an ideal gas undergoing isothermal expansion, the heat input, Q, is exactly the work done by the gas, and is given by Q = nRT ln(V_final/V_initial). This relationship highlights the critical role of heat transfer in maintaining constant temperature during the expansion.

First Law of Thermodynamics

The first law of thermodynamics, essentially a statement of energy conservation, indicates that the change in internal energy of a system is equal to the heat added to the system minus the work done by the system. In mathematical terms, ?U = Q - W, and in processes where the internal energy does not change (such as an isothermal process for an ideal gas), the heat added to the system is equal to the work done by the system.

Work Done in Isothermal Expansion

In an isothermal expansion of an ideal gas, the work is calculated by integrating the pressure with respect to volume. Since the ideal gas law provides a relationship between pressure and volume at a constant temperature, the work done can be found using the integral W = ?(P dV) = nRT ln(V_final/V_initial). This equation is derived under the assumption that temperature remains constant, making the logarithmic dependency on the volume ratio a key feature of the process.

Ideal Gas

An ideal gas is a theoretical model that describes a gas composed of many randomly moving point particles that interact only through elastic collisions. This model simplifies computations by assuming that there are no intermolecular forces and that the volume of the individual molecules is negligible, leading to the ideal gas law, PV = nRT, which relates pressure, volume, temperature, and the amount of gas.

Isothermal Process

An isothermal process is one in which the temperature of the system remains constant throughout the transformation. For an ideal gas, this implies that any change in the internal energy, which depends solely on temperature, is zero. Consequently, any work done by or on the system during an isothermal change must be exactly balanced by heat transfer.

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