Problem 17 - Advanced Problems in Physical Chemistry for Competitive Examinations

Question

The volume of a system becomes twice its original volume on the absorption of 300 cal of heat. The work done on the surrounding was found to be 200 cal. What is \(\Delta U\) for the system? (a) \(500 \mathrm{cal}\) (b) \(300 \mathrm{cal}\) (c) \(100 \mathrm{cal}\) (d) \(-500 \mathrm{cal}\)

Step-by-Step Solution

Answer

The change in internal energy \(\Delta U\) for the system is \(100 \text{cal}\).

Step 1: Understand the First Law of Thermodynamics

The First Law of Thermodynamics states that the change in the internal energy of a system, \(\Delta U\), is equal to the heat, \(Q\), added to the system minus the work, \(W\), done by the system on the surroundings. Mathematically, \(\Delta U = Q - W\).

Step 2: Identify Given Quantities

The heat absorbed by the system is given as \(300 \text{cal}\), and the work done on the surroundings is given as \(200 \text{cal}\). It is important to note that work done on the surroundings is positive which implies the system loses energy.

Step 3: Calculate Change in Internal Energy

Using the formula from the first law, \(\Delta U = Q - W\), we substitute the given values to find \(\Delta U\). Thus, \(\Delta U = 300 \text{cal} - 200 \text{cal} = 100 \text{cal}\).

Key Concepts

Understanding Internal Energy

When discussing thermodynamics, the concept of internal energy is fundamental. It represents the total energy contained within a system, encompassing the kinetic energy of its molecules due to their motion and the potential energy from molecular interactions. This internal energy can change when a system is subjected to heat exchange or when work is performed on or by the system.

For example, when heat is absorbed by a gas, its molecules tend to move faster, which can increase the internal energy. Conversely, when a gas expands and does work on its surroundings, such as pushing a piston, its internal energy can decrease. It's crucial to recognize that while we often discuss heat and work interchangeably with energy, they are not stored within a system like internal energy but are forms of energy transfer.

Thermodynamics – The Science of Heat and Work

The field of thermodynamics is concerned with the movement and conversion of energy, particularly in the form of heat and work. Understanding thermodynamics is key to unlocking how energy is conserved and how it flows from one system to another. The laws of thermodynamics, starting with the first law, provide a framework to predict and explain these energy exchanges in a variety of natural and engineered processes.

Whether it's understanding the efficiency of engines or figuring out the energy cycles within living organisms, thermodynamics touches upon nearly every aspect of science and engineering. The fact that energy cannot be created or destroyed, but only changed from one form to another, is an underlying principle that governs the behavior of all physical systems.

Heat Absorption and Its Effects on a System

The term heat absorption refers to the transfer of thermal energy into a system. This can occur through various mechanisms such as conduction, convection, or radiation. When a system absorbs heat, the added energy can lead to an increase in temperature, a change in state (such as from a solid to a liquid), or an expansion of the system's volume.

It's important to understand that heat is a form of energy transfer resulting from a temperature difference. Thus, when a system, like the one in our exercise example, absorbs heat, we can observe and measure its effects. For instance, as the system absorbs 300 cal, it undergoes a volume expansion, which implies an increase in its internal energy, as indicated by the energy added as heat.

Work Done By and On the System

The concept of work done relates to the force applied over a distance in thermodynamic processes. It is a form of energy transfer that can either increase or decrease a system's internal energy. When a system expands, it pushes against its surroundings and work is done by the system. Conversely, when the surroundings do work on the system - like compressing a gas - this work is considered as being done on the system.

In our textbook exercise, the work done on the surroundings was found to be 200 cal, indicating that the system lost energy as it expanded. This action reduces the internal energy of the system. Therefore, when calculating the change in internal energy, the work done by the system reduces the total energy increase that results from heat absorption. The balance of these energy exchanges is crucial in determining the system's state and behavior after a process has occurred.