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Chapter 2: Problem 51

A gas is contained in a vertical piston-cylinder assembly by a piston weighing \(4,450 \mathrm{~N}\) and having a face area of \(75 \mathrm{~cm}^{2}\) The atmosphere exerts a pressure of \(101.3 \mathrm{kPa}\) on the top of the piston. An electrical resistor transfers energy to the gas in the amount of \(5.3 \mathrm{~kJ}\) as the elevation of the piston increases by \(0.6 \mathrm{~m}\). The piston and cyclinder are poor thermal conductors and friction can be neglected. Determine the change in internal energy of the gas, in \(\mathrm{kJ}\), assuming it is the only significant internal energy change of any component present.

Short Answer

Expert verified
The change in internal energy of the gas is 2.63 kJ.

Step by step solution

01

- Calculate the Work Done by the Piston

The work done by the moving piston can be calculated using the formula: \[ W = F \times d \]Where:- \( W \) is the work done,- \( F \) is the force applied (weight of the piston), and- \( d \) is the displacement.Given:\( F = 4450 \, \text{N} \)\( d = 0.6 \, \text{m} \)Thus:\[ W = 4450 \, \text{N} \times 0.6 \, \text{m} = 2670 \, \text{J} \]
02

- Convert Work Done into kJ

Convert the work done from Joules to kiloJoules: \[ W = 2670 \, \text{J} = 2.67 \, \text{kJ} \]
03

- Calculate Total Energy Transferred to the Gas

The total energy transferred to the gas includes the electrical energy and the work done by the piston. Using the first law of thermodynamics, the change in internal energy can be calculated as follows: \[ \Delta U = Q - W \]Where:- \( \Delta U \) is the change in internal energy,- \( Q \) is the energy transferred to the gas by the electrical resistor, and- \( W \) is the work done by the piston.Given:\( Q = 5.3 \, \text{kJ} \)\( W = 2.67 \, \text{kJ} \)Thus:\[ \Delta U = 5.3 \, \text{kJ} - 2.67 \, \text{kJ} = 2.63 \, \text{kJ} \]

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Key Concepts

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

first law of thermodynamics
The first law of thermodynamics is a cornerstone of thermal physics. It asserts that the energy of a closed system is conserved. This means the total energy change in the system equals the energy added to the system minus the work done by the system on its surroundings.
In mathematical form, it is written as: \(\Delta U = Q - W\), where:
  • \(\Delta U\) is the change in internal energy of the system.
  • \(Q\) represents the heat added to the system.
  • \(W\) is the work done by the system.

When studying a problem involving heat and work, the first law of thermodynamics provides a tool to relate these quantities to changes in internal energy. In our exercise, we see this applied directly to calculate the internal energy change in the gas within the piston-cylinder assembly.
work done by piston
When a piston moves in a cylinder, it performs work on the gas inside. The work done by a piston can be calculated using the formula:
\[W = F \times d\]Here:
  • \(F \) is the force exerted by the piston (its weight, if considering gravity).
  • \(d\) is the displacement or the distance the piston moves.

For this problem, the weight of the piston and the displacement were given, so we found: \[W = 4450 \mathrm{N} \times 0.6 \mathrm{m} = 2670 \mathrm{J}\]
This value was then converted into kilojoules for consistency with other energy units in the problem: \[W = 2.67 \mathrm{kJ}\]
energy transfer
Energy transfer in thermodynamics involves moving energy from one part of the system to another or between the system and its surroundings. In our problem, energy is transferred to the gas in two ways:
  • Through electrical energy from a resistor.
  • Via work done by the piston.

According to our data, the electrical transfer was \[Q = 5.3 \mathrm{kJ}\]
and the work done by the piston was \[W = 2.67 \mathrm{kJ}\]
Using the first law of thermodynamics: \[\Delta U = Q - W \]
  • \(Q \) is the energy transferred to the gas by the resistor.
  • \(W\) is the work done by the piston.

Our calculation leads to an internal energy change of: \[\Delta U = 5.3 \mathrm{kJ} - 2.67 \mathrm{kJ} = 2.63 \mathrm{kJ}\]

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Most popular questions from this chapter

A body is initially at rest then an external force is applied on it which accelerates it with a uniform acceleration of \(1 \mathrm{~m} / \mathrm{s}^{2}\). Mass of the body is \(10 \mathrm{~kg}\). Calculate the work done on the body in \(10 \mathrm{~s}\).

What examples of heat transfer by conduction, radiation, and convection do you encounter when using a charcoal grill?

\(.\) An electric generator supplies electricity to a storage battery at a rate of \(15 \mathrm{~kW}\) for a period of 12 hours. During this \(12-\mathrm{h}\) period there also is heat transfer from the battery to the surroundings at a rate of \(1.5 \mathrm{~kW}\). Then, during the next 12 -h period the battery discharges electricity to an external load at a rate of \(5 \mathrm{~kW}\) while heat transfer from the battery to the surroundings occurs at a rate of \(0.5 \mathrm{~kW}\). (a) For the first 12 -h period, determine, in \(\mathrm{kW}\), the time rate of change of energy stored within the battery. (b) For the second 12 -h period, determine, in \(\mathrm{kW}\), the time rate of change of energy stored in the battery. (c) For the full 24-h period, determine, in \(\mathrm{kJ}\), the overall change in energy stored in the battery.

For a process taking place in a closed system containing gas, the volume and pressure relationship is \(p V^{1.4}=\) constant. The process starts with initial conditions, \(p_{1}=1.5\) bar, \(V_{1}=0.03 \mathrm{~m}^{3}\) and ends with final volume, \(V_{2}=0.05 \mathrm{~m}^{3}\). Determine the work done by the gas.

\(.\) A block of mass \(10 \mathrm{~kg}\) moves along a surface inclined \(30^{\circ}\) relative to the horizontal. The center of gravity of the block. is elevated by \(3.0 \mathrm{~m}\) and the kinetic energy of the block decreases by \(50 \mathrm{~J}\). The block is acted upon by a constant force \(\mathbf{R}\) parallel to the incline, and by the force of gravity. Assume frictionless surfaces and let \(g=9.81 \mathrm{~m} / \mathrm{s}^{2}\). Determine the magnitude and direction of the constant force \(\mathbf{R}\), in \(\mathrm{N}\).
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