Question

A piston-cylinder device contains 5 kg of air at \(400 \mathrm{kPa}\) and \(30^{\circ} \mathrm{C} .\) During a quasi-equilibium isothermal expansion process, \(15 \mathrm{kJ}\) of boundary work is done by the system, and \(3 \mathrm{kJ}\) of paddle-wheel work is done on the system. The heat transfer during this process is \((a) 12 \mathrm{kJ}\) (b) \(18 \mathrm{kJ}\) \((c) 2.4 \mathrm{kJ}\) \((d) 3.5 \mathrm{kJ}\) \((e) 60 \mathrm{kJ}\)

Short answer

Question: A piston-cylinder device contains air initially at 300 K and 200 kPa. During an isothermal process, the air does 15 kJ of boundary work and has 3 kJ of paddle-wheel work done on it. Determine the heat transfer during this process. Answer: (a) 12 kJ

Step-by-step solution

Write the First Law of Thermodynamics equation for an isothermal process

The equation for the first law of thermodynamics for a closed system is: \(\Delta U = Q - W\) where \(\Delta U\) is the change in internal energy, \(Q\) is the net heat transfer, and \(W\) is the work done by the system. For an isothermal process, \(\Delta U = 0\), so the equation becomes: \(Q = W\)

Calculate the total work done by the system during the process

The total work done by the system consists of the boundary work (\(W_b = 15 \,\mathrm{kJ}\)) minus the paddle-wheel work done on the system (\(W_p = 3 \,\mathrm{kJ}\)). Therefore, the total work done by the system is: \(W = W_b - W_p = 15 \,\mathrm{kJ} - 3 \,\mathrm{kJ} = 12 \,\mathrm{kJ}\)

Determine the heat transfer during the process

Now, we can use the first law of thermodynamics for an isothermal process that we found in step 1 to find the heat transfer during the process. Using the total work done by the system calculated in step 2, we have: \(Q = W = 12 \,\mathrm{kJ}\) The heat transfer during this process is 12 kJ, which corresponds to answer choice (a).

Key concepts

Isothermal Process

An isothermal process is a thermodynamic process in which the temperature of the system remains constant. In the context of our exercise, a piston-cylinder device containing air undergoes an isothermal expansion. This implies that the air inside expands at a constant temperature of 30°C. During such a process, the internal energy of an ideal gas remains unchanged because internal energy depends solely on temperature for these types of gases. In practice, maintaining an isothermal condition necessitates heat transfer to or from the system to balance the work done, ensuring the temperature is held steady.

For students grappling with this concept, picture a scenario where the air inside the cylinder is gently and slowly expanding against the piston. Heat is carefully added or removed to compensate for the work done by the expanding gas, in order to keep its temperature from rising or falling. Isothermal processes are often illustrated on a pressure-volume (P-V) diagram as a hyperbolic curve, where they are represented visually by a line that moves horizontally, keeping to a single temperature level.

Internal Energy

Internal energy is a measure of the total energy contained within a system. It is composed of the kinetic and potential energies of the particles within that system. In the context of an isothermal process, the internal energy of an ideal gas does not change because it is a function of temperature. As such, if the temperature does not change, the average kinetic energy of its particles, and hence the internal energy (U), stays the same.

Understanding the concept of internal energy is crucial, especially when tackling thermodynamics problems. The exercise we are analyzing involves an isothermal process, which means that any change in internal energy, denoted as \(\Delta U\), would be zero. This is a key fact that simplifies the application of the first law of thermodynamics, leading us to conclude that any heat added to the system, or removed from it, is directly converted to or from work done by or on the system.

Heat Transfer

Heat transfer is the movement of thermal energy from one object or substance to another as a result of a temperature difference. In the context of thermodynamics exercises and our specific problem, the term 'heat transfer' refers to the thermal energy that crosses the boundary of a system due to the temperature difference between the system and its surroundings.

During an isothermal process, heat transfer is crucial in maintaining constant temperature. It must match the boundary work done by the system to prevent any rise or fall in internal energy. In our piston-cylinder example, heat transfer to the air causes it to expand and do work on the surroundings (piston). The heat transfer value is equal to the net work done, which in the given problem, results in a 12 kJ heat transfer to the system.

Boundary Work

Boundary work occurs when the volume of a gas changes and does work on the environment, or when work is done on the gas, compressing it. In thermodynamic systems like our piston-cylinder device, boundary work is a significant form of energy transfer. It is calculated by integrating force applied over the distance moved by the piston.

In our exercise, the cylinder contains air at a given pressure which expands against the piston, performing boundary work on the surroundings. During the expansion, the volume of the system increases as the piston moves outward. It is important to distinguish this boundary work from other forms of work, such as paddle-wheel work, which are paths dependent and related to processes that stir the system or drive an electrical current. The boundary work is considered path independent in a quasi-static or reversible process. In the given solution, we subtract the paddle-wheel work from the boundary work to find the net work done by the system.