Calculate The Change In Internal Energy (ΔU) Of A Gas That Absorbs 150 J Of Heat And Does 40 J Of Work By Expanding, Using The First Law Of Thermodynamics.

Jun 17, 2025 by ADMIN 156 views

Calculating Change in Internal Energy Using the First Law of Thermodynamics

In the fascinating realm of thermodynamics, we encounter the fundamental laws that govern the transfer of energy and its transformations. One such cornerstone is the First Law of Thermodynamics, a principle that elegantly articulates the conservation of energy. This article delves into a practical application of this law, specifically focusing on calculating the change in internal energy (ΔU) of a gas undergoing a thermodynamic process. We will explore a scenario where a gas absorbs heat from its surroundings and performs work by expanding, providing a clear, step-by-step solution and a comprehensive discussion of the underlying concepts. Understanding the First Law is crucial for anyone delving into physics, chemistry, or engineering, as it provides a framework for analyzing energy changes in various systems.

The First Law of Thermodynamics is essentially a statement of the principle of energy conservation. In simpler terms, it says that energy cannot be created or destroyed, but it can be transferred from one form to another. Mathematically, the First Law is expressed as:

ΔU = Q - W

Where:

ΔU represents the change in internal energy of the system.
Q represents the heat added to the system.
W represents the work done by the system.

Internal energy (ΔU) refers to the total energy contained within a system. This energy encompasses the kinetic and potential energies of the molecules within the system. When heat is added to the system (Q), it increases the internal energy, causing the molecules to move faster and the temperature to rise. Conversely, when the system does work (W), it expends energy, reducing the internal energy. The sign conventions are crucial here: heat absorbed by the system is considered positive (+Q), while heat released by the system is negative (-Q). Similarly, work done by the system is positive (+W), and work done on the system is negative (-W). These conventions ensure that the equation accurately reflects the energy changes occurring in the system. The First Law provides a powerful tool for analyzing thermodynamic processes, allowing us to quantify how energy is exchanged and transformed in various systems, from simple gases to complex engines.

Consider a gas that absorbs 150 J of heat from its surroundings. As it absorbs this heat, the gas expands, performing 40 J of work in the process. Our objective is to calculate the change in internal energy (ΔU) of the gas using the First Law of Thermodynamics. This problem provides a straightforward yet insightful application of the First Law, allowing us to understand how heat and work contribute to changes in a system's internal energy. By carefully considering the sign conventions and applying the formula, we can determine the net change in the energy stored within the gas. This type of calculation is fundamental in thermodynamics and helps us predict and analyze the behavior of various systems under different conditions.

To calculate the change in internal energy (ΔU) of the gas, we will systematically apply the First Law of Thermodynamics. This involves identifying the given values, understanding their signs, and substituting them into the First Law equation. Let's break down the solution step by step:

1. Identify the Given Values

From the problem statement, we have:

Heat absorbed by the gas (Q) = 150 J
Work done by the gas (W) = 40 J

2. Apply Sign Conventions

It is crucial to apply the correct sign conventions to ensure the accuracy of our calculation. According to the conventions:

Heat absorbed by the system is positive, so Q = +150 J.
Work done by the system is positive, so W = +40 J.

3. Use the First Law of Thermodynamics

The First Law of Thermodynamics is given by:

ΔU = Q - W

4. Substitute the Values

Now, we substitute the identified values into the equation:

ΔU = 150 J - 40 J

5. Calculate the Change in Internal Energy

Performing the subtraction, we get:

ΔU = 110 J

Therefore, the change in internal energy (ΔU) of the gas is 110 J. This positive value indicates that the internal energy of the gas has increased. The gas has absorbed more energy as heat than it has expended as work, resulting in a net increase in its internal energy. This step-by-step solution demonstrates a clear application of the First Law, providing a solid understanding of how to approach similar problems in thermodynamics.

To further clarify the calculation, let's reiterate the process with a more detailed explanation of each step. We start with the fundamental equation from the First Law of Thermodynamics:

ΔU = Q - W

Here, ΔU represents the change in the internal energy of the system, Q represents the heat added to the system, and W represents the work done by the system. The critical aspect here is the sign convention. Heat absorbed by the system is considered positive because it increases the system's internal energy. Conversely, heat released by the system is negative, as it decreases the internal energy. Similarly, work done by the system on its surroundings is positive, as it represents energy leaving the system, while work done on the system by the surroundings is negative, representing energy entering the system.

In our specific problem, the gas absorbs 150 J of heat from its surroundings. This means Q = +150 J. The positive sign indicates that energy is entering the system as heat, increasing its internal energy. The gas also does 40 J of work by expanding. This means W = +40 J. Again, the positive sign indicates that the system is doing work on its surroundings, which corresponds to energy leaving the system.

Now we substitute these values into the First Law equation:

ΔU = Q - W

ΔU = (+150 J) - (+40 J)

The equation now reflects the specific energy changes in our system. The gas gains 150 J of energy as heat but loses 40 J of energy as it performs work. To find the net change in internal energy, we perform the subtraction:

ΔU = 150 J - 40 J

ΔU = 110 J

The result, ΔU = 110 J, tells us that the internal energy of the gas has increased by 110 J. The positive sign of ΔU confirms that there is a net increase in the internal energy of the gas. This increase is due to the fact that the gas absorbed more heat than it expended as work. This detailed calculation illustrates the importance of the sign conventions and how they accurately reflect the energy changes within the system. It provides a clear and comprehensive understanding of how the First Law of Thermodynamics can be applied to solve practical problems.

In conclusion, by applying the First Law of Thermodynamics, we successfully calculated the change in internal energy (ΔU) of the gas. The gas absorbed 150 J of heat and performed 40 J of work, resulting in a net increase in internal energy of 110 J. This exercise highlights the fundamental principle of energy conservation and demonstrates how heat and work contribute to the internal energy of a system. Understanding the First Law is essential for analyzing various thermodynamic processes and is a cornerstone in the fields of physics, chemistry, and engineering. By mastering the application of this law, we can better understand and predict the behavior of systems involving energy transfer and transformations. This problem serves as a practical example of how theoretical concepts can be applied to solve real-world scenarios, reinforcing the importance of thermodynamics in scientific and engineering disciplines.