What work will be done, when 3 moles of an ideal gas are...

What work will be done, when 3 moles of an ideal gas are compressed to half the initial volume at a constant temperature of $300 \mathrm{~K}$ ?
(a) $-5188 \mathrm{~J}$
(b) $5000 \mathrm{~J}$
(c) $5188 \mathrm{~J}$
(d) $-5000 \mathrm{~J}$

Key Concepts

Logarithmic Relationships in Thermodynamics

The natural logarithm arises in the equation for work in isothermal processes when integrating the pressure-volume relationship (P = nRT/V). Its use reflects the continuous and exponential nature of volume change in these processes, providing a precise mathematical method to calculate the work done as the gas changes its volume.

Ideal Gas Law

The ideal gas law, expressed as PV = nRT, relates the pressure, volume, temperature, and number of moles of an ideal gas. It provides a simple model that allows us to understand how gases behave under different conditions and forms the foundation for many thermodynamic calculations, including those involving work during gas compression or expansion.

Isothermal Process

An isothermal process is a thermodynamic process in which the temperature remains constant. For an ideal gas undergoing an isothermal process, the internal energy does not change, meaning any work done on or by the gas is exactly balanced by heat transfer. This process is essential in understanding energy exchanges in systems where the temperature is controlled.

Work Done in Isothermal Processes

In an isothermal process for an ideal gas, the work done can be calculated using the equation W = nRT ln(Vf/Vi). This equation is derived by integrating the ideal gas law during a process at constant temperature. The work calculated indicates the energy exchanged through compression or expansion, with work done on the gas being represented as negative when volume decreases.

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