(a) Show that the work done by an ideal gas during the...

(a) Show that the work done by an ideal gas during the quasi-static, isothermal expansion from an initial pressure $P_{i}$ to a final pressure $P_{f}$ is given by $$ W=n R T \ln \frac{P_{f}}{P_{i}} $$ (b) Calculate the work done when the pressure of 1 mol of an ideal gas is decreased quasi-statically from 20 to $1 \mathrm{~atm}$, the temperature remaining constant at $20^{\circ} \mathrm{C}$ $(R=8.31 \mathrm{~J} / \mathrm{mol} \cdot \mathrm{deg})$

(a) Show that the work done by an ideal gas during the quasi-static, isothermal expansion from an initial pressure $P_{i}$ to a final pressure $P_{f}$ is given by
$$
W=n R T \ln \frac{P_{f}}{P_{i}}
$$
(b) Calculate the work done when the pressure of 1 mol of an ideal gas is decreased quasi-statically from 20 to $1 \mathrm{~atm}$, the temperature remaining constant at $20^{\circ} \mathrm{C}$ $(R=8.31 \mathrm{~J} / \mathrm{mol} \cdot \mathrm{deg})$

Key Concepts

Ideal Gas Law

The ideal gas law, PV = nRT, is a fundamental equation in thermodynamics that relates the pressure (P), volume (V), temperature (T), and the number of moles (n) of an ideal gas using the universal gas constant (R). This law provides a simple model for the behavior of gases under many conditions and is essential for deriving other results, such as the work done during different thermodynamic processes.

Isothermal Process

An isothermal process is one in which the temperature remains constant throughout the transformation. For an ideal gas undergoing an isothermal process, the relationship between pressure and volume adjusts in such a way that any change in one variable compensates for the change in the other, while maintaining the product PV constant. This constant temperature condition significantly simplifies the analysis of the work done during expansion or compression.

Thermodynamic Work

In thermodynamics, work is defined as energy transfer resulting from a force acting through a distance. For a gas undergoing a quasi-static expansion or compression, the work done is calculated using the integral W = ?PdV. In the context of an isothermal process for an ideal gas, substituting the expression for pressure from the ideal gas law into the work integral yields a logarithmic form, reflecting how work is accumulated over the expansion or compression.

Integration in Thermodynamics

Integration plays a crucial role in thermodynamics when calculating work done in processes involving continuous changes in state variables. By substituting the state equation (like the ideal gas law) into the integral for work, one can express work in terms of measurable quantities such as pressure and volume. This approach often results in functions such as the natural logarithm, which naturally emerge from integrating reciprocal relationships when temperature is held constant.

Quasi-static Process

A quasi-static process is a thermodynamic process that occurs infinitely slowly, ensuring that the system remains in equilibrium at all stages of the transformation. This idealization permits the use of equilibrium thermodynamics and the application of state equations (like the ideal gas law) continuously along the process path, enabling precise calculations of work and other state functions.

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