Question

A rigid tank initially contains \(3 \mathrm{~kg}\) of air at \(500 \mathrm{kPa}, 290 \mathrm{~K}\). The tank is connected by a valve to a piston-cylinder assembly oriented vertically and containing \(0.05 \mathrm{~m}^{3}\) of air initially at \(200 \mathrm{kPa}, 290 \mathrm{~K}\). Although the valve is closed, a slow leak allows air to flow into the cylinder until the tank pressure falls to \(200 \mathrm{kPa}\). The weight of the piston and the pressure of the atmosphere maintain a constant pressure of \(200 \mathrm{kPa}\) in the cylinder; and owing to heat transfer, the temperature stays constant at \(290 \mathrm{~K}\). For the air, determine the total amount of energy transfer by work and by heat, each in kJ. Assume ideal gas behavior.

Short answer

Work: 10.75 kJ, Heat: 10.75 kJ.

Step-by-step solution

- Understand the problem

Determine the initial and final states of the system - for both the tank and the piston-cylinder assembly. Use the ideal gas law to find the relevant properties.

- Apply the ideal gas law for initial state

For the initial state of the tank, use the ideal gas law:\[ PV = nRT \]Given: \( P_1 = 500 \, \text{kPa} \), \( T_1 = 290 \, \text{K} \), and mass \( m_t = 3 \, \text{kg} \). With air as an ideal gas, use: \( R = 0.287 \, \text{kJ/kg·K} \). Calculate the initial volume of the tank: \[ V_{t1} = \frac{{m_t \, R \, T_1}}{{P_1}} \]

- Calculate the initial number of moles in the cylinder

For the initial state of the cylinder, use the ideal gas law:\[ PV = nRT \]Given: \( P_0 = 200 \, \text{kPa} \), \( T_0 = 290 \, \text{K} \), and volume \( V_{c0} = 0.05 \, \text{m}^3 \). Calculate:\[ n_{c0} = \frac{{P_0 \, V_{c0}}}{{R \, T_0}} \]

- Determine the final state of the system

After the leak, the final pressures in the tank and cylinder are both \( P_2 = 200 \, \text{kPa} \). The temperature remains constant at \( 290 \, \text{K} \). Using the ideal gas law again, calculate the final volume of the air in the tank and the final under constant conditions in the cylinder.

- Energy transfer by work

Work done by the piston-cylinder assembly:\[ W = P \, (V_{c2} - V_{c0}) \]Where \( V_{c2} \) is the final volume, which can be found by the condition that the pressure remains at 200 kPa and temperature stays the same.

- Energy transfer by heat

Use the first law of thermodynamics:\[ \Delta U + W = Q \]Since temperature is constant, internal energy change, \( \Delta U = 0 \):\[ Q = W \]. The heat transfer to maintain isothermal conditions equals the work done.

- Calculate the total work and heat transfer

Substitute in the values calculated for volume change and work done into the equations to find the numerical value. Utilize the initially stated properties and the ideal gas calculations.

Key concepts

Ideal Gas Law

To begin, it is essential to understand the **Ideal Gas Law**, which is represented as \( PV = nRT \). This equation connects the pressure (\(P\)), volume (\(V\)), and temperature (\(T\)) of an ideal gas with the gas constant (\(R\)) and the amount of substance in moles (\(n\)). For air, \(R\) is often given as 0.287 \text{kJ/kg·K\}. In this problem, we'll use the Ideal Gas Law to find the initial and final volumes of air in both a rigid tank and a piston-cylinder assembly respectively, given certain initial pressures, volumes, and temperatures. Remember, the law assumes that the gas behaves ideally, which means it follows the law without deviation.

Piston-Cylinder Assembly

A **piston-cylinder assembly** is a common device in thermodynamics used to study the behavior of gases. It's made up of a cylindrical chamber with a movable piston inside.

In this exercise, the cylinder initially contains air at a certain pressure and volume. The piston can move within the cylinder, allowing for changes in volume while keeping the pressure constant at 200 kPa, thanks to the weight of the piston and atmospheric pressure. This setup enables energy transfer in the form of work and heat. Because the temperature remains constant, it means the process is isothermal.

Energy Transfer Calculations

Energy transfer in thermodynamics can occur via work or heat. Here, **work (\(W\))** done by the system is calculated in the piston-cylinder assembly. When the piston moves, it performs work on the surroundings. The work formula for a constant pressure process is \( W = P \times (V_{c2} - V_{c0}) \), where \(V_{c2}\) and \(V_{c0}\) are the final and initial volumes, respectively.

**Heat (\(Q\))** transfer, however, is determined through the First Law of Thermodynamics. If the temperature remains constant (\(\text{isothermal process}\)), any work done must be balanced by heat transfer to maintain the internal energy \( \text{(ΔU)}\) as zero. Hence, \( Q = W \).

Rigid Tank Analysis

The **rigid tank** in this exercise is crucial for understanding the overall behavior of the system. A rigid tank has a fixed volume and doesn't change when pressure or temperature changes. Initially, this tank contains 3 kg of air at 500 kPa and 290 K. The mass stays constant while the gas leaks.

We need to use the Ideal Gas Law to calculate the initial and final states.
Initial state: \( P_1 = 500 \text{ kPa}, T_1 = 290 \text{ K}, \text{ and } m_t = 3 \text{ kg} \).
Final state: \( P_2 = 200 \text{ kPa}, T_2 = 290 \text{ K} \).
The calculations for initial and final states help determine the volume changes and understand the dynamics of gas movement into the piston-cylinder assembly.

First Law of Thermodynamics

The **First Law of Thermodynamics** is a fundamental concept in energy transfer. It states that the change in internal energy of a system is equal to the heat added to the system minus the work done by the system on its surroundings. Mathematically, it's expressed as: \[ ΔU + W = Q \].

In this problem, since the temperature remains constant, the change in internal energy (\( ΔU \)) is zero. Therefore, \[ Q = W \], meaning the heat transferred to maintain isothermal conditions equals the work done by the system. This principle helps solve for the total amount of energy in kJ transferred by work and heat, ensuring the understanding of how these energy exchanges influence the system.