Problem 23 - University Physics with Modern Physics

Question

Heat \(Q\) flows into a monatomic ideal gas, and the volume increases while the pressure is kept constant. What fraction of the heat energy is used to do the expansion work of the gas?

Step-by-Step Solution

Answer

The fraction of heat energy used for work is \(\frac{2}{5}\).

Step 1: Understanding the Problem

We need to find the fraction of heat energy (\(Q\)) used to do work (\(W\)) when a monatomic ideal gas expands at constant pressure.

Step 2: Remembering Key Formulas

For a process at constant pressure, the first law of thermodynamics states: \( Q = \Delta U + W \), where \(\Delta U\) is the change in internal energy and \(W\) is the work done by the system.

Step 3: Determining Change in Internal Energy

For a monatomic ideal gas, the change in internal energy is \(\Delta U = \frac{3}{2}nR\Delta T\). Since pressure is constant, \(W = P\Delta V\). Also, \(W = nR\Delta T\), given that \(P\Delta V = nR\Delta T\).

Step 4: Calculating Fraction of Heat Doing Work

The fraction of heat used for work is \( \frac{W}{Q} = \frac{nR\Delta T}{Q} \). Substitute \(Q = \Delta U + W = \frac{3}{2}nR\Delta T + nR\Delta T\).

Step 5: Simplifying the Fraction

The total heat added is \(Q = \frac{5}{2}nR\Delta T \). Hence, the fraction of heat energy used to do work is \( \frac{W}{Q} = \frac{nR\Delta T}{\frac{5}{2}nR\Delta T} = \frac{2}{5}\).

Key Concepts

Monatomic Ideal Gas

In thermodynamics, a monatomic ideal gas is a fundamental concept for understanding the behavior of gases under various conditions. These gases consist of single-atom particles, meaning their simplicity makes it easier to predict and calculate their responses to changes in temperature, pressure, and volume.
A key feature of a monatomic ideal gas is that its internal energy extit{U} depends only on its temperature and the amount of substance extit{n}. The formula relating the internal energy to temperature is given by:

• \( U = \frac{3}{2}nRT \)

where \(R\) is the universal gas constant and \(T\) is the temperature in Kelvin. This equation highlights why monatomic gases are excellent models for studying basic thermodynamic principles. Knowing how changes in these conditions affect the gas, we can derive other important quantities, such as work done and heat absorbed.
When we apply this knowledge, it helps simplify the calculations in exercises involving monatomic ideal gases, particularly when exploring their heat and expansion characteristics.

First Law of Thermodynamics

The First Law of Thermodynamics is crucial in understanding energy transformations in any thermodynamic system, including monatomic ideal gases. This law, which is essentially the law of energy conservation, states that the energy added to the system as heat must equal the change in internal energy plus the work done by the system on its surroundings.

• Mathematically expressed as: \( Q = \Delta U + W \)

where:

• \(Q\) is the heat added to the system
• \(\Delta U\) is the change in internal energy
• \(W\) is the work done by the system

Understanding this law allows you to track how energy flows in and out of the system. It helps relate how much of the energy is retained as internal heat and how much is used to perform work, such as when gas expands at constant pressure. These principles are at the heart of comprehending the behavior of gases in thermodynamic processes.

Heat Energy

In thermodynamics, heat energy \(Q\) plays a critical role in changing the state of a substance. When heat energy is added to a monatomic ideal gas, it can either increase the gas's internal energy or perform expansion work.

• For a process at constant pressure, the heat energy is divided between changing the internal energy and expansion work: \( Q = \frac{3}{2}nR\Delta T + nR\Delta T \)

Here, each term represents:

• \(\frac{3}{2}nR\Delta T\) indicating the change in internal energy
• \(nR\Delta T\) describing the work done by the system

This energy distribution reflects the very essence of thermodynamic cycles, where a portion of input energy is invested in altering the internal energy, while the rest is allocated to performing work. Understanding this balance is key to mastering concepts like efficiencies of engines and refrigerators, which rely on transforming heat energy into useful work.

Expansion Work

Expansion work is a concept in thermodynamics describing the work a system does as it expands against an external pressure. For a monatomic ideal gas expanding at constant pressure, the work done during expansion can be calculated by:

• \( W = P\Delta V \)
• Alternatively, equivalently as: \( W = nR\Delta T \)

These equations translate the expansion process into understandable math, showcasing how changes in volume relate to energy. In our context, the exercise determined the fraction of heat energy used to do expansion work, found by the equation:

• \( \frac{W}{Q} = \frac{2}{5} \)

This fraction means that 40% of the heat energy supplied is used to perform work during the expansion. Recognizing how much energy is used this way gives insights into the efficiency and functioning of gas processes. It also emphasizes the importance of understanding thermodynamic principles when analyzing energy transformations in practical applications.