10 mole of ideal gas expand isothermally and reversibly from a pressure of \(10 \mathrm{~atm}\) to \(1 \mathrm{~atm}\) at \(300 \mathrm{~K}\). What is the largest mass which can lifted through a height of 100 meter? (a) \(31842 \mathrm{~kg}\) (b) \(58.55 \mathrm{~kg}\) (c) \(342.58 \mathrm{~kg}\) (d) None of these

The concept of work done by a gas is central to understanding several physical processes, including the operation of heat engines and refrigerators. During expansion, a gas can do work against an external pressure. This work is essentially the energy transferred by the gas to its surroundings. In thermodynamics, the work done by a gas during an expansion or compression is a critical parameter.

In an isothermal reversible expansion, such as the exercise mentioned, the temperature of the gas remains constant, which implies that the internal energy of the gas doesn't change, and the work done is equal to the heat absorbed by the system from the surroundings. The formula for the work done in an isothermal process utilizes the natural logarithm of the volume ratio and is given by:
\[W = nRT \times \text{ln}\left(\frac{V_f}{V_i}\right)\]
where \(n\) is the number of moles of gas, \(R\) is the ideal gas constant, \(T\) is the absolute temperature, \(V_f\) is the final volume, and \(V_i\) is the initial volume. For an ideal gas undergoing an isothermal expansion, this formula allows us to calculate the work done by the gas as it expands against an external pressure.

The Ideal Gas Law is a fundamental principle in chemistry and physics that describes the behavior of an ideal gas. An ideal gas is a hypothetical gas that perfectly follows the relationship between pressure, volume, temperature, and the number of moles of gas. Although no real gas is truly 'ideal', many come close under a range of conditions, making the ideal gas law very useful. The law is mathematically represented as:
\[PV = nRT\]
where \(P\) stands for pressure, \(V\) is volume, \(n\) is the number of moles, \(R\) is the universal gas constant, and \(T\) is the absolute temperature in Kelvin. This law combines several simpler gas laws, such as Boyle's Law, Charles's Law, and Avogadro's Law, into one comprehensive equation. In the context of the exercise provided, the ideal gas law helps to connect the initial and final states of the gas during its isothermal expansion.

Boyle's Law is a gas law which states that the pressure and volume of a gas are inversely proportional to each other at a constant temperature for a fixed mass of an ideal gas. In other words, if the volume of a gas increases, the pressure decreases, provided the temperature and the number of moles of gas remain constant, and vice versa. This relationship can be mathematically expressed as:
\[P_iV_i = P_fV_f\]
where \(P_i\) and \(P_f\) are the initial and final pressures, and \(V_i\) and \(V_f\) are the initial and final volumes of the gas, respectively. Boyle's Law is particularly relevant in the exercise because it allows us to relate the initial state to the final state of the system during the isothermal process, which is necessary to calculate the work done by the expanding gas.

Gravitational potential energy is the energy stored in an object due to its position in a gravitational field. More simply, it is the energy that an object has because of its height above the ground. When you lift an object against the force of gravity, you’re increasing its gravitational potential energy. The formula to calculate this energy for an object of mass \(m\) lifted to a height \(h\) in the Earth's gravity is given by:
\[E_p = mgh\]
where \(g\) represents the acceleration due to gravity, which is approximately \(9.81 \text{ m/s}^2\) on Earth. This concept is especially useful in relating the work done by the gas in the provided exercise to the physical task of lifting a mass through a certain height. Through this comparison, students can better visualize the amount of work that the gas can perform as a tangible measure, like the weight of an object being lifted, enriching their understanding of energy conversion and conservation principles.