Chapter 5: Problem 14
In a gas expansion, \(87 \mathrm{~J}\) of heat is released to the surroundings and the energy of the system decreases by \(128 \mathrm{~J}\). Calculate the work done.
Short Answer
Expert verified
The work done by the system is 41 J.
Step by step solution
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Internal Energy
Internal energy forms the cornerstone of understanding thermodynamic systems. It refers to the total energy contained within a system, stemming from the kinetic and potential energies of the molecules that constitute it. In simple terms, it's like the fuel available to do work or produce heat. Internal energy changes when added heat or mechanical work alter the motion and interactions between particles.
Expressed mathematically, the change in internal energy (\( \Delta U \)) is a fundamental concept that connects heat exchange and work done. When a system expands or contracts, the internal energy will change depending on how much heat is absorbed or released and the work done by or on the system. The equation \( \Delta U = Q - W \) illustrates this beautifully, balancing the internal energy against heat (\( Q \)) and work (\( W \)).
Always remember that an increase in internal energy can mean the system has absorbed heat or work has been done on it, while a decrease usually implies heat release or work done by the system.
Expressed mathematically, the change in internal energy (\( \Delta U \)) is a fundamental concept that connects heat exchange and work done. When a system expands or contracts, the internal energy will change depending on how much heat is absorbed or released and the work done by or on the system. The equation \( \Delta U = Q - W \) illustrates this beautifully, balancing the internal energy against heat (\( Q \)) and work (\( W \)).
Always remember that an increase in internal energy can mean the system has absorbed heat or work has been done on it, while a decrease usually implies heat release or work done by the system.
Heat Transfer
Heat transfer is the process by which thermal energy flows from a hotter object to a cooler one. This movement of heat can happen through conduction, convection, or radiation. In thermodynamics, heat is a transfer of energy due to temperature difference, and it plays a major role in determining the net internal energy of a system.
In the exercise, the system releases \( 87 \, \mathrm{J} \) of heat to the surroundings, indicating that it is losing energy to its environment. When heat flows out, as in this case, the heat (\( Q \)) is considered negative. This helps us measure the actual energy change in the system. Understanding the direction of heat transfer is crucial, as it helps predict how the system's internal energy will alter and, subsequently, how much work can be done.
In the exercise, the system releases \( 87 \, \mathrm{J} \) of heat to the surroundings, indicating that it is losing energy to its environment. When heat flows out, as in this case, the heat (\( Q \)) is considered negative. This helps us measure the actual energy change in the system. Understanding the direction of heat transfer is crucial, as it helps predict how the system's internal energy will alter and, subsequently, how much work can be done.
Work Done
The concept of work done in thermodynamics describes the energy transferred by a force acting over a distance. When a thermodynamic system expands or contracts, it performs work on its surroundings, usually against external pressure. Work done by the system is conventionally considered positive, indicating energy is exerted outwards. Conversely, when the system contracts due to external work done on it, it is negative.
In this example, calculations showed the system performed \( 41 \, \mathrm{J} \) of work. This positive value directly tells us that the system is expanding, doing work on its surroundings. The concept ties back to the First Law of Thermodynamics, where the equation \( \Delta U = Q - W \) captures this interplay. Recognizing the sign (positive or negative) for work done helps in grasping whether energy is entering or leaving the system.
In this example, calculations showed the system performed \( 41 \, \mathrm{J} \) of work. This positive value directly tells us that the system is expanding, doing work on its surroundings. The concept ties back to the First Law of Thermodynamics, where the equation \( \Delta U = Q - W \) captures this interplay. Recognizing the sign (positive or negative) for work done helps in grasping whether energy is entering or leaving the system.
Energy Conservation
The principle of energy conservation is central to the First Law of Thermodynamics, stating that energy cannot be created or destroyed, only transformed from one type into another. Within any closed system, the total energy remains constant. This means any change in a system's internal energy must equal the net heat exchanged minus the work done by or on the system.
In the exercise, the decrease in internal energy \( -128 \, \mathrm{J} \) combined with the heat released \( -87 \, \mathrm{J} \) leads directly to understanding the conservation of energy. The calculated work, \( 41 \, \mathrm{J} \), shows how energy rearranges without any loss or gain of total system energy. Energy conservation emphasizes that while forms of energy can change, such as from thermal to mechanical, the total energy remains unified in balance.
In the exercise, the decrease in internal energy \( -128 \, \mathrm{J} \) combined with the heat released \( -87 \, \mathrm{J} \) leads directly to understanding the conservation of energy. The calculated work, \( 41 \, \mathrm{J} \), shows how energy rearranges without any loss or gain of total system energy. Energy conservation emphasizes that while forms of energy can change, such as from thermal to mechanical, the total energy remains unified in balance.
