How much work is done by an ideal gas in expanding...

How much work is done by an ideal gas in expanding isothermally from an initial volume of $3.00$ liters at $20.0$ atm to a final volume of $24.0$ liters? For an isothermal expansion by an ideal gas, $$ \begin{aligned} \Delta W &=P_{1} V_{1} \ln \left(\frac{V_{2}}{V_{1}}\right) \\ &=\left(20.0 \times 1.01 \times 10^{5} \mathrm{~N} / \mathrm{m}^{2}\right)\left(3.00 \times 10^{-3} \mathrm{~m}^{3}\right) \ln \left(\frac{24.0}{3.00}\right)=12.6 \mathrm{~kJ} \end{aligned} $$

How much work is done by an ideal gas in expanding isothermally from an initial volume of $3.00$ liters at $20.0$ atm to a final volume of $24.0$ liters?
For an isothermal expansion by an ideal gas,
$$
\begin{aligned}
\Delta W &=P_{1} V_{1} \ln \left(\frac{V_{2}}{V_{1}}\right) \\
&=\left(20.0 \times 1.01 \times 10^{5} \mathrm{~N} / \mathrm{m}^{2}\right)\left(3.00 \times 10^{-3} \mathrm{~m}^{3}\right) \ln \left(\frac{24.0}{3.00}\right)=12.6 \mathrm{~kJ}
\end{aligned}
$$

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Key Concepts

Logarithmic Relationship in Isothermal Work Calculation

When an ideal gas expands isothermally, the work done is obtained by integrating the expression derived from the ideal gas law, resulting in a natural logarithm function of the volume ratio. This logarithmic relationship, expressed as ln(V?/V?), is crucial for computing the work performed during isothermal changes.

Ideal Gas Behavior

The behavior of an ideal gas is described by the ideal gas law (PV = nRT), which relates pressure, volume, and temperature. In an isothermal process, this law implies that the product of pressure and volume remains constant, simplifying the integration used for calculating work done during expansion or compression.

Work in Thermodynamic Processes

In thermodynamics, the work done by a system during expansion or compression is calculated as the integral of pressure with respect to volume (?PdV). This concept is central to understanding energy transfer processes and is used to derive the work expression for various thermodynamic pathways.

Isothermal Process

An isothermal process is a thermodynamic process in which the temperature of the system remains constant. For an ideal gas undergoing an isothermal change, the internal energy remains unchanged, implying that any heat added to the system is directly used to perform work during expansion or compression.

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