Question

A piston-cylinder device initially contains \(1.2 \mathrm{kg}\) of air at \(700 \mathrm{kPa}\) and \(200^{\circ} \mathrm{C}\). At this state, the piston is touching on a pair of stops. The mass of the piston is such that 600 -kPa pressure is required to move it. A valve at the bottom of the tank is opened, and air is withdrawn from the cylinder. The valve is closed when the volume of the cylinder decreases to 80 percent of the initial volume. If it is estimated that \(40 \mathrm{kJ}\) of heat is lost from the cylinder, determine \((a)\) the final temperature of the air in the cylinder, (b) the amount of mass that has escaped from the cylinder, and \((c)\) the work done. Use constant specific heats at the average temperature.

Short answer

Based on the given information and the step by step solution provided, calculate the final temperature of air in the cylinder, the amount of mass that escaped, and the work done during this process.

Step-by-step solution

Calculate initial state of air

First, we will determine the initial properties of the air in the piston-cylinder device. We are given that initially, \(m_1 = 1.2\mathrm{kg}\), \(P_1 = 700\mathrm{kPa}\) and \(T_1 = 200^{\circ}\mathrm{C}\). Let's calculate the initial specific volume \(v_1\) using the ideal gas law: $$ Pv = mRT $$ Where \(R\) is the specific gas constant of air \(= 287\,\mathrm{J/kg\cdot K}\). Therefore, $$ v_1 = \frac{m_1RT_1}{P_1} $$

Calculate final properties

Next, we will determine the final properties of the air in the piston-cylinder device. We are given that the final volume \(V_2\) is 80% of the initial volume \(V_1\). So, $$ v_2 = 0.8v_1 $$ Also, we know that the pressure required to move the piston is \(600\,\mathrm{kPa}\), so the final pressure will be: $$ P_2 = P_1 - 600\,\mathrm{kPa} $$

Apply the first law of thermodynamics

Now, we will apply the first law of thermodynamics to find the final temperature. The first law can be written as: $$ Q - W = m(u_2 - u_1) $$ We are given that \(Q = -40\,\mathrm{kJ}\). We will estimate the work done during the process, \(W\), using the formula: $$ W = P_2(V_2 - V_1) $$ To find the internal energy difference, we can write: $$ u_2 - u_1 = c_v(T_2 - T_1) $$ Where \(c_v\) is the specific heat at constant volume. We are given to use constant specific heats at the average temperature. Using the first law and our derived expressions, we can calculate the final temperature \(T_2\).

Calculate the mass that escaped

To find the mass of air that has escaped, we can use the ideal gas law for the final state: $$ m_2 = \frac{P_2V_2}{RT_2} $$ Then, the mass that escaped is: $$ \Delta m = m_1 - m_2 $$

Calculate the work done

Lastly, we will use the expression we derived earlier to calculate the work done during the process: $$ W = P_2(V_2 - V_1) $$ Using these steps, we can find the final temperature of the air in the cylinder, the amount of mass that has escaped from the cylinder, and the work done during the process.