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When a gas expands against an external pressure, it performs work by pushing the surroundings. The amount of work done by the gas during this expansion can be determined using the formula:

• \[W = -P_{\text{external}} \cdot \Delta V\]

This formula calculates the work done when a gas volume changes (\(\Delta V\)) against a constant external pressure (\(P_{\text{external}}\)). The negative sign indicates that the work done by the system is energy transferred out of the system.
In the case of expansion, the final volume (\(V_f\)) is greater than the initial volume (\(V_i\)), meaning \(\Delta V\) is positive, resulting in a negative work value. This signifies that the gas has done work and expended energy. The work done during expansion can be used, for instance, to lift a weight to a certain height.

The compression process is essentially the opposite of expansion. Here, work is done on the gas by the surroundings, increasing the internal energy of the system. During compression, the work done on the gas is calculated similarly to expansion, but in this scenario, it is positive:

• \[W' = P_i \cdot \Delta V\]

This indicates that energy is being added to the gas. The initial pressure (\(P_i\)) and the change in volume (\(\Delta V\)) are used in this formula, where \(\Delta V\) is still the final volume minus the initial volume (\(V_f - V_i\)).
During compression, \(\Delta V\) is often negative, leading to a positive work value. This means energy is entering the system, often causing temperature or pressure increases due to the decreased volume and increased density of molecules.

The ideal gas law is a fundamental equation that describes the behavior of ideal gases. It relates pressure, volume, temperature, and the number of moles of gas:

• \[PV = nRT\]

Here, \(P\) is the pressure, \(V\) is the volume, \(n\) is the number of moles, \(R\) is the universal gas constant, and \(T\) is the temperature.
Using this law, one can calculate the initial and final volumes of gas during a process under constant temperature conditions, as shown in the exercise.
For example, to find the initial volume (\(V_i\)), rearrange the law to:

• \[V_i = \frac{n \cdot R \cdot T}{P_i}\]

Similarly, the final volume (\(V_f\)) can be found using:

• \[V_f = \frac{n \cdot R \cdot T}{P_f}\]

This helps in calculating the change in volume (\(\Delta V\)), an essential step in determining the work done by or on the gas.

The mass calculation in this scenario is concerned with determining the maximum weight that the gas can lift during expansion, and the minimum weight that can restore the system during compression. This involves using the work done and converting it into lifting potential:

• \[W = m \cdot g \cdot h\]

Here, \(m\) is the mass, \(g\) is the acceleration due to gravity, and \(h\) is the height through which the mass is lifted. By rearranging the formula, you can find the mass:

• \[m = \frac{W}{g \cdot h}\]

This formula shows that the work done on the mass depends not only on the work done by the gas but also on the gravitational force and the height. Similarly, during compression, the minimal mass required to perform the work on the gas can be found using:

• \[m' = \frac{W'}{g \cdot h}\]

In this way, by knowing the calculated work values from the expansion or compression, one can find the masses \(m\) and \(m'\) responsible for lifting or being lifted under these thermodynamic conditions.