91Ó°ÊÓ

The ideal gas law is a fundamental equation that describes the behavior of an ideal gas. This equation is represented as \( PV = nRT \), where \( P \) is the pressure of the gas, \( V \) is the volume, \( n \) is the number of moles of gas, \( R \) is the universal gas constant, and \( T \) is the temperature in kelvins. In simple terms, the law connects the pressure, volume, and temperature of a fixed quantity of gas, assuming there are no interactions between the gas particles and they occupy no space. This idealization makes it easier to predict the behavior of gases under different conditions.

When solving problems like the one at hand, we use the ideal gas law to calculate unknown properties of a gas when the other properties are provided. As seen in the problem, understanding how to manipulate the formula to solve for the desired variable is key to finding solutions in exercises involving changes in pressure, volume or temperature of an ideal gas.

Isothermal compression refers to the process of reducing the volume of a gas while maintaining its temperature constant. As the ‘iso’ prefix suggests, 'the same', isothermal processes occur at a consistent temperature. This is in contrast to adiabatic processes, where no heat is exchanged with the environment.

In isothermal compression of an ideal gas, since the temperature remains constant, according to the ideal gas law \( PV = nRT \), the product of pressure \( P \) and volume \( V \) remains constant throughout. If the volume decreases, the pressure must increase proportionally, and vice versa. This behavior is encapsulated in Boyle's Law, which is a special case of the ideal gas law for isothermal conditions. This concept aids in understanding the thermodynamic processes and solving problems such as calculating final volumes after a compression, as exemplified in the exercise.

When we talk about work done by or on a gas in thermodynamics, we refer to the energy transferred by the gas as it expands or by the surroundings as the gas is compressed. The formula to calculate the work done during an isothermal process is \( W = -nRT \ln\left(\frac{V_{f}}{V_{i}}\right) \), where \( n \) is the number of moles, \( R \) is the universal gas constant, \( T \) is the temperature, \( V_{i} \) is the initial volume, and \( V_{f} \) is the final volume.

The minus sign indicates that work is done on the gas when \( V_{f} < V_{i} \), as in compression, and done by the gas during expansion \( V_{f} > V_{i} \). For instance, in the given problem, the work was calculated based on the natural logarithm of the ratio of the initial and final volumes. This calculation determines how much energy is needed to compress the gas at a constant temperature.

The first law of thermodynamics, also known as the law of energy conservation, states that energy cannot be created or destroyed in an isolated system. Instead, it can only be transformed from one form to another. Mathematically, it is expressed as \( \Delta U = Q - W \), where \( \Delta U \) is the change in the internal energy of the system, \( Q \) is the heat added to the system, and \( W \) is the work done by the system.

In an isothermal process for an ideal gas, the internal energy is only a function of temperature. Therefore, if the temperature remains constant, there is no change in internal energy (\( \Delta U = 0 \)). Consequently, any work done on the gas (such as during compression) must be balanced by an equal amount of heat transferred out of the gas, or vice versa. This principle enabled the solution of part (c) in the given exercise, illustrating the direct relationship between heat transfer and work in thermodynamic processes.