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Chapter 3: Problem 120

Nitrogen \(\left(\mathrm{N}_{2}\right)\) fills a closed, rigid tank fitted with a paddle wheel, initially at \(540^{\circ} \mathrm{R}, 20 \mathrm{lbf} / \mathrm{in}^{2}\), and a volume of \(2 \mathrm{ft}^{3}\). The gas is stirred until its temperature is \(760^{\circ} \mathrm{R}\). During this process, heat transfer from the gas to its surroundings occurs in an amount \(1.6\) Btu. Assuming ideal gas behavior, determine the mass of the nitrogen, in lb, and the work, in Btu. Kinetic and potential energy effects can be ignored.

Short Answer

Expert verified
The mass of the nitrogen is calculated using the ideal gas law. The work done by the paddle wheel is found by applying the first law of thermodynamics.

Step by step solution

01

Convert initial temperature

Convert the initial temperature from Rankine to Fahrenheit to use in later calculations. Remember, the Rankine scale is given by \[ T_F = T_R - 459.67 \].
02

Apply Ideal Gas Law to Find Mass

Use the ideal gas law to find the mass of nitrogen: \[ PV = mRT \]. Given \[ P = 20 \text{ lbf/in}^2 \], \[ V = 2 \text{ ft}^3 \], \[ R = 55.165 \text{ ft} \text{ lb} / (\text{lb} \text{ mol} \text{ R}) \], and \[ T_i = 540 \text{ R} \]. The molecular weight of nitrogen, \[ MW = 28.014 \text{ lb} / (\text{lb mol}) \]. Solve for the mass, \[ m \].
03

Determine Final Conditions

Given the final temperature \[ T_f = 760 \text{ R} \], use the ideal gas law again to confirm that the volume remains constant and validate any further conditions for the work calculations.
04

Calculate Change in Internal Energy

Calculate the change in internal energy, \[ \triangle U = mC_v(T_f - T_i) \]. For nitrogen, \[ C_v = 0.171 \text{ Btu}/(\text{lb} \text{ R}) \].
05

Apply the First Law of Thermodynamics

Use the first law of thermodynamics, \[ \triangle U = Q - W \], to solve for the work done by the paddle wheel. Given \[ Q = -1.6 \text{ Btu} \] and the previously found \[ \triangle U \], solve for \[ W \].

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Thermodynamics
Thermodynamics is the branch of physics that deals with heat, work, and the forms of energy involved in physical and chemical processes. It outlines how energy is transferred and transformed. Basic principles include the interaction between work and heat, and laws that govern these processes.
Thermodynamics can be divided into four laws: Zeroth, First, Second, and Third. The primary focus for this exercise is the First Law, which deals with energy conservation. We'll also encounter tools like the ideal gas law, which helps us analyze behavior under varying conditions.
Internal Energy Change
Internal energy is the total energy contained within a system due to microscopic motion and interactions of molecules. It comprises kinetic energy (due to molecule movement) and potential energy (due to molecular forces).
For gases, internal energy change (abla U) can be calculated when temperature changes. For the nitrogen gas in this problem, using the specific heat at constant volume (abla C_v), we use the formula:
abla U = mC_v(T_f - T_i)
where:
  • m = mass
  • C_v = specific heat at constant volume
  • T_i = initial temperature
  • T_f = final temperature
Understanding this helps in determining how much heat energy is converted into internal energy with a given heat transfer.
Ideal Gas Law
The ideal gas law combines various gas properties into one equation:
PV = nRT
where:
  • P = pressure
  • V = volume
  • n = number of moles of the gas
  • R = universal gas constant
  • T = temperature
For an ideal gas, it explains the relationship between pressure, volume, temperature, and amount of gas. When dealing with a rigid tank like in our problem, the volume (V) remains constant.
To find the mass (m) of nitrogen, we use:
m = (PV) / (RT) * MW
where MW is the molecular weight of nitrogen.
The law ensures that we can find unknown properties when temperature or pressure changes.
First Law of Thermodynamics
The First Law of Thermodynamics states that energy within an isolated system is conserved: it cannot be created or destroyed, only transferred or converted from one form to another. Mathematically, it's expressed as:
abla U = Q - W
where:
  • abla U = change in internal energy
  • Q = heat added to the system
  • W = work done by the system
In our problem, with known heat transfer (Q = -1.6 Btu) and calculated internal energy change (abla U), we determine the work done (W) by rearranging the equation:
W = Q - abla U
Thus, the First Law provides a straightforward approach to understanding the energy shifts in our nitrogen-filled tank.

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Most popular questions from this chapter

A closed, rigid tank fitted with a paddle wheel contains \(0.1 \mathrm{~kg}\) of air, initially at \(300 \mathrm{~K}, 0.1 \mathrm{MPa}\). The paddle wheel stirs the air for 20 minutes, with the power input varying with time according to \(\dot{W}=-10 t\), where \(\dot{W}\) is in watts and \(t\) is time, in minutes. The final temperature of the air is \(1060 \mathrm{~K}\). Assuming ideal gas behavior and no change in kinetic or potential energy, determine for the air (a) the final pressure, in MPa, (b) the work, in kJ, and (c) the heat transfer, in kJ.

Air contained in a piston-cylinder assembly, initially at 2 bar, \(200 \mathrm{~K}\), and a volume of \(1 \mathrm{~L}\), undergoes a process to a final state where the pressure is 8 bar and the volume is \(2 \mathrm{~L}\). During the process, the pressure-volume relationship is linear. Assuming the ideal gas model for the air, determine the work and heat transfer, each in kJ.

A closed, rigid tank fitted with a paddle wheel contains \(2 \mathrm{~kg}\) of air, initially at \(300 \mathrm{~K}\). During an interval of 5 minutes, the paddle wheel transfers energy to the air at a rate of \(1 \mathrm{~kW}\). During this interval, the air also receives energy by heat transfer at a rate of \(0.5 \mathrm{~kW}\). These are the only energy transfers. Assuming the ideal gas model for the air, and no overall changes in kinetic or potential energy, determine the final temperature of the air, in \(\mathrm{K}\).

A system consisting of \(1 \mathrm{~kg}\) of \(\mathrm{H}_{2} \mathrm{O}\) undergoes a power cycle composed of the following processes: Process 1-2: Constant-volume heating from \(p_{1}=5\) bar, \(T_{1}=\) \(160^{\circ} \mathrm{C}\) to \(p_{2}=10\) bar. Process 2-3: Constant-pressure cooling to saturated vapor. Process 3-4: Constant-volume cooling to \(T_{4}=160^{\circ} \mathrm{C}\). Process 4-1: Isothermal expansion with \(Q_{41}=815.8 \mathrm{~kJ}\). Sketch the cycle on \(T-v\) and \(p-v\) diagrams. Neglecting kinetic and potential energy effects, determine the thermal efficiency.

A well-insulated, rigid tank contains \(1.5 \mathrm{~kg}\) of Refrigerant \(134 \mathrm{~A}\), initially a two-phase liquid-vapor mixture with a quality of \(60 \%\) and a temperature of \(0^{\circ} \mathrm{C}\). An electrical resistor transfers energy to the contents of the tank at a rate of \(2 \mathrm{~kW}\) until the tank contains only saturated vapor. For the refrigerant, locate the initial and final states on a \(T-v\) diagram and determine the time it takes, in \(s\), for the process.
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