Question

Air is contained in a variable-load piston-cylinder device equipped with a paddle wheel. Initially, air is at \(400 \mathrm{kPa}\) and \(17^{\circ} \mathrm{C} .\) The paddle wheel is now turned by an external electric motor until \(75 \mathrm{kJ} / \mathrm{kg}\) of work has been transferred to air. During this process, heat is transferred to maintain a constant air temperature while allowing the gas volume to triple. Calculate the required amount of heat transfer, in kJ/kg.

Short answer

Answer: The amount of heat transfer is 75 kJ/kg.

Step-by-step solution

Gather the information given

Initial pressure of air (P1) is 400 kPa. Initial temperature of air (T1) is 17°C = 290K (converting to Kelvin by adding 273). The volume of the gas triples (V2 = 3V1). Work transferred to air (W) is 75 kJ/kg.

Use the ideal gas law to find the Volume ratio

Let's assume air behaves as an ideal gas. So, we can use the ideal gas equation, given by: \(PV = nRT\) Since T1 = T2 (At constant temperature) and m (mass) is constant, R (Gas constant) is also constant. Therefore, we can rewrite the volume ratio of the cylinder as: \(V_1=\frac{n R T_1}{P_1}\) \(V_2=\frac{n R T_2}{P_2}\) Divide both equations: \(\frac{V_2}{V_1} = \frac{P_1 * T_2}{P_2 *T_1}\) As we have that \(V_2 =3V_1\). Thus, we must find \(P_2\)

Determine the final pressure

Since the volume tripled and the temperature is constant, we can use the volume ratio to find the final pressure (P2): \(\frac{3V_1}{V_1} = \frac{P_1 * T_1}{P_2 *T_1}\) Canceling out \(T_1\) and \(V_1\) terms and solve for P2: \(P_2 = \frac{P_1}{3}\) Plug in the initial pressure (P1): \(P_2 = \frac{400}{3} \approx 133.33 \mathrm{kPa}\)

Determine the work done on the gas

We are given the work done by the paddle wheel on the gas (W) = 75 kJ/kg. We need the work done by the gas (W_gas): \(W_{gas}=-W= -75 \mathrm{kJ/kg}\)

Use the first law of thermodynamics to find the heat transfer

The first law of thermodynamics states that the change in internal energy (ΔU) is equal to the heat added (Q) minus the work done by the gas (W_gas): \(ΔU = Q - W_{gas}\) However, since the temperature of the gas is constant, the change in internal energy is zero (ΔU = 0). \(0 = Q - (-75)\) To find the heat transfer (Q), we need to add the value of work done by the gas to both sides of the equation: \(Q = 75 \mathrm{kJ/kg}\) So, the required amount of heat transfer is 75 kJ/kg.

Key concepts

First Law of Thermodynamics

At its core, the First Law of Thermodynamics deals with the notion of energy conservation, encapsulating the principle that energy can neither be created nor destroyed—only transformed from one form to another. This fundamental concept is pivotal in heat transfer calculations.

In the context of our exercise, where a paddle wheel works on a piston-cylinder device containing air, this law translates to the understanding that the work done on the air inside the cylinder will affect the internal energy and, by extension, may require heat transfer to maintain a constant temperature.

The formula that succinctly represents this law is:
\[\Delta U = Q - W\]
where \(\Delta U\) is the change in internal energy, \(Q\) is the heat added to the system, and \(W\) is the work done by the system. In cases like our exercise where temperature—and therefore internal energy—remains constant, \(\Delta U\) equals zero, simplifying the relationship to \(Q = W\), which enables us to directly determine the heat required to keep the temperature unchanged during the work process.

Ideal Gas Law

The Ideal Gas Law is a cornerstone in understanding and calculating the behavior of gases under various conditions of pressure, volume, and temperature. Expressed by the equation
\[PV = nRT\],
it relates pressure (\(P\)), volume (\(V\)), amount of substance (\(n\) in moles), the ideal gas constant (\(R\)), and temperature (\(T\)) in Kelvin.

In our example, the Ideal Gas Law is instrumental for deriving a relationship between the initial and final states of the air in the cylinder. Since the process maintains constant temperature--a key stipulation of our problem—the law simplifies to \(P_1V_1 = P_2V_2\), showing that the product of pressure and volume initially is equal to the product of pressure and volume after the process, when air behaves as an ideal gas.

When dealing with heat transfer calculations, utilizing the Ideal Gas Law allows us to predict how a gas will respond to external work, contributing to the overall energy management within a system. It's this predictive capability that enables us to determine the final pressure after the volume of the gas has changed, which is a critical step in resolving our heat transfer calculation.

Work-Energy Principle

Closely tied to the First Law of Thermodynamics is the Work-Energy Principle. This principle fundamentally asserts that the work done on an object is equal to the change in energy of that object. When it comes to gases, this work can induce changes in the internal energy which can be observed as changes in temperature or phase, under the assumption that no energy is lost.

In the exercise, the paddle wheel introduces work (\(W\)) to the air in the piston-cylinder device. This work could potentially alter the internal energy of the air, which, following the Work-Energy Principle, would usually lead to a change in temperature or state. However, to keep the air temperature constant, the energy inputted by the paddle wheel (representing work done on the gas) must be offset by heat transfer (\(Q\)).

This illustrates a practical application of the Work-Energy Principle in thermodynamics: the work done by the external motor via the paddle wheel is counterbalanced by the heat exchanged with the surroundings, in order to maintain the air temperature and meet the conditions defined in the exercise.