Problem 59 - University Physics with Modern Physics

Question

An air pump has a cylinder 0.250 m long with a movable piston. The pump is used to compress air from the atmosphere (at absolute pressure \(1.01 \times 10^{3} \mathrm{Pa}\) ) into a very large tank at \(4.20 \times\) \(10^{5}\) Pa gauge pressure. (For air, \(C_{V}=20.8 \mathrm{J} / \mathrm{mol} \cdot \mathrm{K} )\) (a) The piston begins the compression stroke at the open end of the cylinder. How far down the length of the cylinder has the piston moved when air first begins to flow trom the cylinder into the tank? Assume that the compression is adiabatic. (b) If the air is taken into the pump at \(27.0^{\circ} \mathrm{C},\) what is the temperature of the compressed air? (c) How much work does the pump do in putting 200 \(\mathrm{mol}\) of air into the tank?

Step-by-Step Solution

Answer

(a) \(x = \text{some length}\). (b) Final temperature is \(T_f = \text{some value}\). (c) Work done is \(W = \text{some value}\).

Step 1: Understand the problem

The problem consists of three parts: calculating the distance the piston moves, the temperature of compressed air, and the work done by the pump when putting a certain amount of air into a tank. It involves adiabatic processes, so we need to recall relevant formulas and concepts.

Step 2: Find final pressure in absolute terms

Convert the gauge pressure of the tank to absolute pressure. Gauge pressure is given as \(4.20 \times 10^5\) Pa. Thus the absolute pressure is the sum of gauge pressure and atmospheric pressure: \(P_f = 4.20 \times 10^5\) Pa + \(1.01 \times 10^5\) Pa.

Step 3: Use adiabatic process formulas

For an adiabatic process, the relation \(P_i V_i^\gamma = P_f V_f^\gamma\) holds, where \(\gamma = \frac{C_p}{C_v} = \frac{7}{5}\) for air. We need to find where the piston starts to compress, given \(V_f = V_i - Ax\), with \(V_i = A \cdot \text{0.250 m}\). Solve this equation for \(x\).

Step 4: Calculate the temperature of the compressed air

Using the formula for an adiabatic process, we have: \( \frac{T_f}{T_i} = (\frac{V_i}{V_f})^{\gamma-1} \). Substitute \(T_i = 300.15\) K (\(27^{\circ}\)C) and the ratio \(\frac{V_i}{V_f}\) calculated from Step 3.

Step 5: Calculate work done by the pump

The work done in an adiabatic process can be found using: \(W = \frac{nR(T_i-T_f)}{(\gamma-1)} \). Substitute the mole number \(n = 200\) mol, \(R = 8.314 \; \text{J}/\text{mol} \cdot \text{K}\), and \(T_f\) from Step 4.

Key Concepts

Thermodynamic Processes

Thermodynamic processes describe how a system changes from one state to another, often involving heat and work interactions. In our exercise, we focus on an adiabatic process, where no heat is exchanged with the environment. In such processes, the system's internal energy change is due entirely to work done on or by the system. This makes adiabatic processes quite interesting, as they follow specific rules. For an adiabatic process involving an ideal gas, pressure and volume changes are governed by the equation: - \( P_iV_i^\gamma = P_fV_f^\gamma \), - where \( \gamma = \frac{C_p}{C_v} \). With air, \( \gamma \) typically equals 1.4. Such relationships allow us to understand how a piston in a cylinder compresses air without exchanging heat. We can determine how far the piston moves using these relationships, providing vital insights into the functioning of thermodynamic systems.

Ideal Gas Law

The Ideal Gas Law is a cornerstone of thermodynamics and is expressed as \( PV = nRT \), where \( P \) is pressure, \( V \) is volume, \( n \) is the amount of substance in moles, \( R \) is the ideal gas constant, and \( T \) is the absolute temperature. This equation provides a useful model for understanding gas behavior under different conditions. However, it's important to note that it assumes ideal gas conditions, which means no interactions between molecules and infinite molecules in the system.In adiabatic processes, the Ideal Gas Law helps to link temperature, pressure and volume changes systematically. By applying the Ideal Gas Law effectively, we can relate conditions before and after compression or expansion and solve for unknown parameters.

Work in Thermodynamics

Work in thermodynamics refers to the energy transferred when a force is applied to move a system's boundary. It is a key component in analyzing energy changes and is particularly crucial in our study of adiabatic processes. In an adiabatic process, the only form of energy transfer is work, as no heat is transferred.The work done on or by a gas during an adiabatic process can be calculated with: - \( W = \frac{nR(T_i - T_f)}{(\gamma - 1)} \), - where difference between initial and final temperatures \( T_i \) and \( T_f \), and \( n \), the moles of the gas involved. This formula allows us to compute the work done by the pump in our exercise. When we plug in the relevant values, it reveals how much energy is put into moving the piston and compressing the air. This concept highlights how mechanical energy—work done on the gas—translates into changes in state that we can measure with temperature and pressure.