Question

A spring-loaded piston-cylinder device contains \(1 \mathrm{kg}\) of carbon dioxide. This system is heated from 100 kPa and \(25^{\circ} \mathrm{C}\) to \(1000 \mathrm{kPa}\) and \(300^{\circ} \mathrm{C} .\) Determine the total heat transfer to and work produced by this system.

Short answer

Answer: The total heat transfer to the system is 279.94 kJ, and the work produced by the system is 50.69 kJ.

Step-by-step solution

Calculate the initial specific volume

We are given the initial pressure, temperature, and mass of the carbon dioxide. We can use the ideal gas law to find the initial specific volume (marked as \(v_1\)). The specific gas constant for carbon dioxide, \(R_{CO_2}\), is approximately \(0.1889 \frac{kJ}{kg \cdot K}\). The ideal gas law for this problem can be written as: $$ PV = mRT $$ Solving for the initial specific volume: $$ v_1 = \frac{RT_1}{P_1} $$ Given the initial pressure, \(P_1 = 100\,\text{kPa}\), and initial temperature, \(T_1 = 25^{\circ} \text{C} = 298.15\,\text{K}\), we can calculate \(v_1\): $$ v_1 = \frac{0.1889 \cdot 298.15}{100} = 0.05632 \frac{m^3}{kg} $$

Calculate the specific internal energy at each state

Using thermodynamic tables for carbon dioxide, we can find the specific internal energy (\(u_1\) and \(u_2\)) at the initial and final state. For \(P_1\) and \(T_1\), we have: $$ u_1 = 427.19\,\frac{kJ}{kg} $$ For the final pressure, \(P_2 = 1000\,\text{kPa}\), and final temperature, \(T_2 = 300^{\circ} \text{C} = 573.15\,\text{K}\): $$ u_2 = 656.44\,\frac{kJ}{kg} $$

Write the first law of thermodynamics for the system

The first law of thermodynamics for the system can be written as: $$ Q - W = m(u_2 - u_1) $$ Where \(Q\) is the total heat transfer to the system, \(W\) is the work produced by the system, and \(m\) is the mass of the carbon dioxide.

Calculate the work produced by the system

We can calculate the work produced by using the formula: $$ W = m \int_{P_1}^{P_2} v \, dP $$ Since we are dealing with ideal gas behavior and there is no change in the specific volume during the process (as per given 'spring-loaded piston' condition), we can calculate the work done as follows: $$ W = m \cdot v_1(P_2 - P_1) $$ Substituting the numerical values: $$ W = 1 \cdot 0.05632 \cdot (1000 - 100) = 50.688 \,kJ $$

Calculate the total heat transfer to the system

Now we can use the first law of thermodynamics to find the total heat transfer, \(Q\): $$ Q = W + m(u_2 - u_1) $$ Substituting the numerical values: $$ Q = 50.688 + 1 \cdot (656.44 - 427.19) = 279.938 \, kJ $$ So, the total heat transfer to the system is \(279.94\, kJ\) (rounded to two decimal places), and the work produced by the system is \(50.69\, kJ\) (rounded to two decimal places).

Key concepts

Specific Volume

Understanding the 'specific volume' is crucial for comprehending a variety of thermodynamic processes. Specific volume, denoted as 'v', is simply the volume occupied by a unit mass of a substance.

In the realm of gases, where we often assume ideal behavior, this property is directly related to the temperature and pressure of the gas, as encapsulated by the Ideal Gas Law: \(PV = mRT\). Here, 'P' represents pressure, 'V' is volume, 'm' is mass, 'R' is the specific gas constant, and 'T' is temperature. Therefore, for a fixed mass of gas, specific volume can be calculated using \(v = \frac{V}{m}\), with V being the volume of the gas.

In our textbook exercise, once we know the pressure and temperature of carbon dioxide, we can rearrange the Ideal Gas Law to solve for specific volume \(v_1\) at a given state. This specific volume will then be a crucial parameter for determining other thermodynamic properties of the system, such as work produced.

Internal Energy

The 'internal energy' of a system, typically represented as 'U' or 'u' for specific internal energy, is directly tied to the kinetic and potential energies of the molecules within the system. For many applications, we consider only specific internal energy, which provides a measure of energy per unit mass.

In practical terms, specific internal energy is often determined using thermodynamic tables, as it depends on properties like temperature and pressure. By consulting these tables for carbon dioxide at different states, as demonstrated in the exercise, one can find the internal energy before and after a process. When a substance undergoes a thermodynamic process, the change in internal energy (\(u_2 - u_1\)) is associated with heat transfer and work done, serving as a central component of the first law of thermodynamics.

Heat Transfer

The concept of 'heat transfer' describes the movement of thermal energy from one thing to another due to a temperature difference. In thermodynamics, this transfer is denoted by the letter 'Q' and can occur in various ways, namely conduction, convection, and radiation.

Heat can be either added to or removed from a system, resulting in a change in the system's internal energy or facilitating the system's ability to do work. In our textbook exercise, when the piston-cylinder device containing carbon dioxide is heated, it experiences a net heat transfer into the system. Using the first law of thermodynamics, we discern that this added heat results in an increase in internal energy and the production of work, as the carbon dioxide expands.

Work Produced

In thermodynamics, 'work' is energy transferred by a system to its surroundings, conventionally defined when the system expands or contracts against external forces. The work produced by a system can be symbolized by 'W', and in the context of a gas within a piston, it is typically calculated by integrating pressure over the change in volume.

The exact calculation of work depends on the process path. For an ideal gas in a piston, if specific volume remains constant due to the presence of a 'spring-loaded piston' as mentioned in our exercise, work can be simplified to \(W = m \(v_1\)(P_2 - P_1)\). This distinct relationship between pressure, volume, and work is pivotal for engineers and scientists in the design and analysis of engines, turbines, and other mechanical systems where compressible fluids, like gases, are involved.