The work done in an adiabatic change of fixed amount of an ideal gas depends on change in (a) volume (b) pressure (c) temperature (d) density

In thermodynamics, an adiabatic process is one where no heat transfer occurs between the system and its surroundings. One primary characteristic of such a process is that all energy transferred to or from the system does the work or changes the internal energy of the system.

For an ideal gas expanding or compressing adiabatically, the work done (\( W \) is associated with the change in volume but is most fundamentally related to the change in temperature. Why? Because as the temperature of an ideal gas changes, its internal energy also changes, which accounts for the work done. Hence, while volume and pressure do alter during the process, they are not independent variables—they are interconnected with temperature changes. For example, if a gas is adiabatically compressed, its internal energy increases leading to a rise in temperature, which in turn affects pressure and volume.

The first law of thermodynamics is essentially a statement of energy conservation, and it plays a central role in understanding adiabatic processes. This law states that the change in internal energy (\( \triangle U \) of a system is equal to the heat added to the system (\( Q \) minus the work done by the system (\( W \) on the surroundings. Mathematically, it's expressed as:\[ \triangle U = Q - W \].

In an adiabatic process, since no heat is transferred (\( Q = 0 \) the first law simplifies to \( \triangle U = -W \), where work done by the system is considered positive. Therefore, the work done in an adiabatic process is equivalent to the change in internal energy of the system, and since we're often dealing with ideal gases, this internal energy change can be further linked to the change in temperature.

Internal energy is a measure of a system's total energy, encompassing both the random kinetic energies of its molecules and any potential energies they may have due to their positions or arrangements. In terms of ideal gases, internal energy is a function of temperature, such that \( \triangle U = nC_v\triangle T \), where \( n \) is the number of moles, \( C_v \) is the molar specific heat at constant volume, and \( \triangle T \) is the change in temperature.

The reason temperature is central in this equation is that for an ideal gas, its internal energy doesn't depend on pressure or volume but solely on temperature—a higher temperature means higher average kinetic energy of the gas particles, which equates to a higher internal energy.

The ideal gas law states that \( PV = nRT \), where \( P \) represents pressure, \( V \) volume, \( n \) the amount of substance in moles, \( R \) the ideal gas constant, and \( T \) temperature. This law provides a simple model of gas behavior under a variety of conditions, assuming that the molecules have no volume and no intermolecular forces.

In relation to an adiabatic process, the behavior of an ideal gas is predictable in terms of pressure, volume, and temperature changes. The specific relationship between these variables during an adiabatic process is given by Poisson's Law, which is \( PV^\tau = \text{constant} \), where \( \tau \) is the heat capacity ratio (\( C_p / C_v \) of the gas. This law helps explain why, even as an ideal gas expands and its volume increases, its temperature can decrease (due to no heat transfer to maintain temperature), causing a drop in pressure.