A When an ideal gas is compressed at constant temperature...

A When an ideal gas is compressed at constant temperature (isothermal conditions), $\Delta E$ is zero. Consider $9.00 \mathrm{~L}$ of a gas that is initially at $1.00 \mathrm{~atm}$ and $25^{\circ} \mathrm{C}$. (a) Calculate $q$ and $w$ if the gas sample is compressed isothermally at a constant pressure of 9.00 atm to a final volume of $1.00 \mathrm{~L}$. (b) Calculate $q$ and $w$ if the gas is compressed isothermally in two steps: first at a constant pressure of $3.00 \mathrm{~atm}$ (to an intermediate volume of $3.00 \mathrm{~L}$ ) and then at a pressure of $9.00 \mathrm{~atm}$ (and a final volume of 1.00 L). (c) Calculate $q$ and $w$ of a three-step isothermal compression (first to $2.00 \mathrm{~atm}$ and $4.50 \mathrm{~L},$ then to $3.00 \mathrm{~atm}$ and $3.00 \mathrm{~L},$ and finally $9.00 \mathrm{~atm}$ and $1.00 \mathrm{~L}$ ). Compare with the two- and one-step compressions. (d) Based on your results for parts a to c, suggest how this compression might be carried out so that the minimum amount of work would be done on the system.

A When an ideal gas is compressed at constant temperature (isothermal conditions), $\Delta E$ is zero. Consider $9.00 \mathrm{~L}$ of a gas that is initially at $1.00 \mathrm{~atm}$ and $25^{\circ} \mathrm{C}$.
(a) Calculate $q$ and $w$ if the gas sample is compressed isothermally at a constant pressure of 9.00 atm to a final volume of $1.00 \mathrm{~L}$.
(b) Calculate $q$ and $w$ if the gas is compressed isothermally in two steps: first at a constant pressure of $3.00 \mathrm{~atm}$ (to an intermediate volume of $3.00 \mathrm{~L}$ ) and then at a pressure of $9.00 \mathrm{~atm}$ (and a final volume of 1.00 L).
(c) Calculate $q$ and $w$ of a three-step isothermal compression (first to $2.00 \mathrm{~atm}$ and $4.50 \mathrm{~L},$ then to $3.00 \mathrm{~atm}$ and $3.00 \mathrm{~L},$ and finally $9.00 \mathrm{~atm}$ and $1.00 \mathrm{~L}$ ). Compare with the two- and one-step compressions.
(d) Based on your results for parts a to c, suggest how this compression might be carried out so that the minimum amount of work would be done on the system.

Key Concepts

Ideal Gas

An ideal gas is a theoretical model where the particles do not interact and occupy negligible volume, obeying the ideal gas law (PV = nRT). This law relates pressure, volume, and temperature, providing a simple framework for predicting and analyzing the behavior of gases under various conditions.

Isothermal Process

An isothermal process is a thermodynamic process that occurs at a constant temperature. For an ideal gas undergoing such a process, the internal energy remains constant, meaning there is no change in temperature-dependent energy. This property simplifies the analysis since the heat exchanged (q) is exactly equal in magnitude and opposite in sign to the work done (w) on or by the system.

First Law of Thermodynamics

The first law of thermodynamics is the principle of energy conservation applied to thermodynamic systems. It states that the change in internal energy (?E) of a system is equal to the sum of heat (q) added to the system and the work (w) done on the system. In an isothermal process for an ideal gas, because ?E is zero, the heat absorbed or released is directly balanced by the work done, making it a cornerstone for analyzing energy transfer during these processes.

Work in Thermodynamic Processes

Work in the context of thermodynamics is defined as the energy transferred by a system when its volume changes under pressure. For an ideal gas, the work done on or by the system during a reversible isothermal process can be calculated using the integral of pressure with respect to volume, often expressed as nRT ln(Vf/Vi). This work is path dependent, meaning that the total work done depends on the process path taken between the initial and final states.

Path Dependence in Thermodynamics

Path dependence in thermodynamics refers to the fact that while state functions such as internal energy depend only on the initial and final states, quantities like work and heat depend on the specific process or path taken between those states. This concept is crucial when considering multistep processes versus a single-step process for an isothermal compression, as the division of the process into steps at different pressures can change the work required, even though the initial and final states are identical.

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