91Ó°ÊÓ

When talking about the work done on a gas, we're diving into how energy is transferred during expansion or compression. In this exercise, we saw that the gas expanded from an initial state \(V_i\) to a final state \(3V_i\) and a pressure change from \(P_i\) to \(3P_i\). This expansion is done in such a manner that pressure is directly proportional to volume.

To determine the work done, we use the equation:

• \( W = \int_{V_{i}}^{3V_{i}} P \, dV \)

Given this proportionality (\(P = kV\)), when we integrate from \(V_i\) to \(3V_i\), we find the amount of work done.

Inserting the relationship into the integral, we perform the calculation:

• \( W = \int_{V_{i}}^{3V_{i}} kV \, dV = k \left[ \frac{V^2}{2} \right]_{V_i}^{3V_i} = 3kV_i^2 = 3P_iV_i \)

This reflects the energy used (or work done) in expanding the gas within these constraints. The work done is the area under the curve in a Pressure-Volume graph, equivalent to energy transferred.

The temperature-volume relationship in this context is central to our understanding of gas behavior under the conditions set by the problem. By using the Ideal Gas Law, \(PV = nRT\), we explore how temperature varies with volume.

Given that pressure is proportional to volume (\(P = kV\)), substituting into the ideal gas equation gives:

• \(kV^2 = nRT\)
• Reorganizing for temperature gives: \(T = \frac{kV^2}{nR}\)

This formula tells us that temperature is a function of the square of the volume, illustrating how temperature changes with respect to volume.

During the process described, as the gas volume increases from \(V_i\) to \(3V_i\), the temperature also varies accordingly due to this quadratic relationship. This means that temperature changes in a nonlinear manner with volume, which is key to predicting gas behavior under such conditions.

Direct proportionality between pressure and volume, \(P = kV\), is a unique scenario within gas dynamics. Often we expect pressure to decrease as volume increases, but here, they increase together.

In this exercise, we see that as the gas expands from \(V_i\) to \(3V_i\), the pressure transitions similarly from \(P_i\) to \(3P_i\). This steady proportional growth is indicative of a controlled environment where energy input is constant throughout the process.

• Understanding this relationship is key: as the volume triples, so too does the pressure.
• This becomes crucial in systems where predictable pressure-volume behavior is needed, such as in hydraulics or controlled laboratory settings.

The beauty of this term in the Ideal Gas Law context is its simplification of calculations related to work done and temperature changes, allowing us deeper insights with more straightforward computation.