Question

A mass of \(1.5 \mathrm{kg}\) of air at \(120 \mathrm{kPa}\) and \(24^{\circ} \mathrm{C}\) is contained in a gas-tight, frictionless piston-cylinder device. The air is now compressed to a final pressure of 600 kPa. During the process, heat is transferred from the air such that the temperature inside the cylinder remains constant. Calculate the work input during this process.

Short answer

Given: - mass of air (m) = 1.5 kg - initial pressure (P1) = 120,000 Pa - temperature (T) = 297.15 K - specific gas constant (R) = 287 J/(kg·K)

Step-by-step solution

List given data and find the initial volume

From the problem, we have: - mass of air (m) = 1.5 kg - initial pressure (P1) = 120 kPa = 120,000 Pa - final pressure (P2) = 600 kPa = 600,000 Pa - temperature (T) = 24°C = 24 + 273.15 = 297.15 K We also know that air can be treated as an ideal gas. For air, R (specific gas constant) = 287 J/(kg·K). Using the ideal gas law, we can find the initial volume (V1): PV = mRT => V1 = mRT / P1

Key concepts

Ideal Gas Law

The ideal gas law is a fundamental equation in thermodynamics that relates the pressure, volume, and temperature of an ideal gas with the amount of substance present. Mathematically represented as PV=nRT, it consolidates Boyle's Law, Charles's Law, and Avogadro's Law. However, in contexts involving mass instead of the number of moles, it’s more convenient to use the form: \( PV = mRT \), where 'P' is pressure, 'V' is volume, 'm' is mass, 'R' is the specific gas constant, and 'T' is temperature in Kelvin.

In our exercise, air is assumed to behave as an ideal gas, which is a good approximation under normal conditions. The specific gas constant for air (R) is given as 287 J/(kg·K). By using the given mass, temperature, and initial pressure, the initial volume \( V1 \) can be calculated as \( V1 = \frac{mRT}{P1} \), allowing us to move on to the next step of finding the work done during the isothermal compression process.

Work Done During Compression

Work done during compression of a gas is an important concept in thermodynamics. It refers to the energy required to reduce a gas's volume against its pressure. For isothermal processes, where temperature remains constant, the work done \( W \) on an ideal gas can be determined using the formula \( W = nRT \ln\left(\frac{V2}{V1}\right) \) with variables representing the number of moles \( n \), gas constant \( R \), initial and final volumes \( V1 \) and \( V2 \), and temperature \( T \). However, for calculations using mass, we use \( W = mRT \ln\left(\frac{V2}{V1}\right) \).

It's essential to remember that because compression reduces volume, work done on the gas is positive, reflecting the energy input into the system. In the given exercise, once the initial volume is determined, the final volume can be found from the final pressure (assuming constant temperature), and the work done on the gas can be calculated.

Thermodynamics

Thermodynamics is the branch of physics that deals with the relationships between heat and other forms of energy. In essence, it's concerned with energy conversion and the laws that govern such processes. In the context of our exercise, the thermodynamic process in question is isothermal compression — a process that occurs under constant temperature.

The first law of thermodynamics, also known as the law of energy conservation, states that energy cannot be created or destroyed in an isolated system. This principle implies that the work input during the compression of the gas must equal the energy transferred from the system in the form of heat, minus any changes in internal energy. Since the temperature is constant, there's no change in internal energy for an ideal gas, making the work done equal to the heat transferred out of the gas.

Understanding these fundamental principles provides the tools necessary to approach and solve complex problems in thermodynamics and is crucial for students to grasp to move beyond just solving the problem to understanding the why and how behind the solutions.