What will be the work done when one mole of a gas expands isothermally from \(15 \mathrm{~L}\) to \(50 \mathrm{~L}\) against a constant pressure of 1 atm at \(25^{\circ} \mathrm{C}\) ? (a) \(-3542 \mathrm{cal}\) (b) \(-843.3 \mathrm{cal}\) (c) \(-718 \mathrm{cl}\) (d) \(-60.23 \mathrm{cal}\)

Understanding the pressure-volume work formula is paramount when it comes to assessing the labor a system experiences during processes such as expansion or compression of gases. The core formula is given by the equation: \[ W = -P \Delta V \] where:

• \( W \) represents the work done on or by the system,
• \( P \) stands for the constant pressure the system is under, and
• \( \Delta V \) denotes the change in volume, which is found by subtracting the initial volume (\( V_i \)) from the final volume (\( V_f \)).

In thermodynamics, if the process is isothermal, meaning the temperature remains constant, gases can do work. The negative sign indicates that work is done by the system when the gas expands; conversely, work is done on the system during compression. Applying this concept to an isothermal expansion of a gas at a constant pressure means the environment loses energy equal to the work done by the system.

Unit conversion is a fundamental concept to comprehend in physics and engineering, crucial for ensuring consistency and accuracy in calculations. It's essential to work within the International System of Units (SI) to keep your mathematics flawless. For instance, volumes are often given in liters (L) but must be converted to cubic meters (\( m^3 \)) using the conversion factor of 1 L = 0.001 \( m^3 \).
Similarly, when dealing with pressure, standard atmospheric pressure (\( 1 \) atm) can be converted to Pascals (\( Pa \)) using the conversion 1 atm = 101,325 \( Pa \).
For energy, Joules (\( J \)) are the SI units, where 1 calorie (\( cal \)) equals 4.184 Joules. Mastery of these conversions is crucial for accurate application of the pressure-volume work formula and for the correct interpretation of results in terms of the units commonly used in the study of thermodynamics.

The ideal gas law is a cornerstone of thermodynamics and gas physics, quantitatively describing the relationship between the four state variables—pressure (\( P \)), volume (\( V \)), temperature (\( T \)), and the number of moles (\( n \))—for an ideal gas. Expressed by the equation: \[ PV = nRT \]

• \( P \) is the pressure,
• \( V \) is the volume,
• \( n \) is the number of moles of the gas,
• \( R \) is the ideal gas constant, and
• \( T \) is the absolute temperature in Kelvins (\( K \)).

This law denotes that for any quantity of gas at a constant temperature and pressure, the volume is proportional to the number of moles. However, the ideal gas law is an approximation and works best under conditions of low pressure and high temperature. In the context of the exercise, knowing the number of moles and temperature allows us to estimate the work done during the gas expansion if the ideal gas behavior is assumed.

Thermodynamics is a branch of physics that deals with the relationships between heat and other forms of energy. In particular, it studies the conversion of energy into work and vice versa, governed by four fundamental laws. For an isothermal process, where the temperature stays constant, the internal energy of an ideal gas remains unchanged. This implies that any heat added to the system does work but does not change the energy within the system.
Understanding thermodynamics is indispensable when analyzing processes like isothermal gas expansion. It aids in comprehending concepts like work, internal energy, and heat transfer. In the provided exercise, isothermal expansion under a constant external pressure can be examined using concepts from thermodynamics. The work is calculated via the pressure-volume work formula and considering that the temperature is held constant, the energy transfer is understood better in terms of work done by the gas.