Problem 31 - University Physics with Modern Physics

Question

Two moles of carbon monoxide (CO) start at a pressure of 1.2 atm and a volume of 30 liters. The gas is then compressed adiabatically to \(\frac{1}{3}\) this volume. Assume that the gas may be treated as ideal. What is the change in the internal energy of the gas? Does the internal energy increase or decrease? Does the temperature of the gas increase or decrease during this process? Explain.

Step-by-Step Solution

Answer

The internal energy and temperature of the gas increase during the adiabatic compression.

Step 1: Understand the Adiabatic Compression

In an adiabatic compression, no heat is exchanged with the surroundings. For an ideal gas, the change in internal energy (\[\Delta U\]) is equal to the work done on the gas, since \(\Delta U = q + w\), where \(q = 0\) in adiabatic processes. Thus, the change in internal energy equals the work done on the gas.

Step 2: Apply the First Law of Thermodynamics

In an adiabatic process, \(q = 0\), so the first law of thermodynamics simplifies to \(\Delta U = w\). Since work is done on the gas during compression, the internal energy will increase.

Step 3: Use the Adiabatic Process Relationship

For an adiabatic process, the relation between initial and final states of an ideal gas is given by \(P_1 V_1^\gamma = P_2 V_2^\gamma\), where \(\gamma = \frac{C_p}{C_v}\). For diatomic gases like CO, \(\gamma \approx 1.4\).

Step 4: Calculate Final Pressure Using Adiabatic Relation

Given \(V_2 = \frac{1}{3} V_1\), substitute into the adiabatic relation: \[ P_1 V_1^\gamma = P_2 (\frac{1}{3} V_1)^\gamma \]Solve for \(P_2\):\[ P_2 = P_1 \left(3\right)^\gamma \]\(P_2 = 1.2 \text{ atm} \times 3^{1.4}\).

Step 5: Calculate Change in Temperature

Use the relation for temperatures in adiabatic processes: \(T_1 V_1^{\gamma-1} = T_2 V_2^{\gamma-1}\). Since \(V_2 = \frac{1}{3}V_1\), solve for \(T_2\):\[ T_2 = T_1 \left(3\right)^{\gamma-1} \]. Elementary manipulation shows \(T_2 > T_1\), indicating an increase in temperature.

Step 6: Conclusion about Internal Energy and Temperature Change

Since there is work done on the gas during compression, as indicated by the increasing temperature, the internal energy of the gas increases. Therefore, temperature and internal energy both increase during adiabatic compression of an ideal gas.

Key Concepts

Ideal Gas Law

The Ideal Gas Law is a fundamental principle for understanding gas behaviors in different conditions. It is represented by the equation \( PV = nRT \), where:

• \( P \) is the pressure of the gas
• \( V \) is the volume it occupies
• \( n \) is the number of moles of the gas
• \( R \) is the ideal gas constant
• \( T \) is the temperature in Kelvin

This equation helps correlate the pressure, volume, and temperature of a gas under a given set of conditions. It assumes the gas behaves ideally, which means the gas particles do not attract or repel each other, and occupy no volume. In the context of the adiabatic process, the Ideal Gas Law is indirectly used to understand changes in these variables as the gas volume changes.
By knowing how these factors interact, we can predict how a gas will behave if it is compressed or expanded, such as in an adiabatic process.

Internal Energy

Internal energy is the total energy contained within a system, derived from the motion and interactions of the molecules within the gas. In ideal gases, internal energy is primarily dependent on the temperature of the gas and the number of particles it contains. Internal energy can be calculated using the equation:\[ \Delta U = nC_v\Delta T \]
Where \( \Delta U \) is the change in internal energy, \( n \) is the number of moles, \( C_v \) is the molar specific heat capacity at constant volume, and \( \Delta T \) is the change in temperature.
During an adiabatic process, since no heat is exchanged with the surroundings, any change in the internal energy of an ideal gas is due to the work done on or by the gas. Therefore, in an adiabatic compression, the work done on the gas increases its internal energy and temperature.

First Law of Thermodynamics

The First Law of Thermodynamics is a version of the law of conservation of energy, tailored for thermodynamic systems and processes. It is typically stated as:\[\Delta U = q + w\]Where:

• \( \Delta U \) is the change in internal energy of the system
• \( q \) is the heat added to the system
• \( w \) is the work done on the system

For adiabatic processes, this law simplifies because no heat exchange occurs with the surroundings \((q = 0)\). Hence, it reduces to \( \Delta U = w \), meaning the entire change in internal energy stems from the work done on the gas.
In the context of the given problem, understanding this law helps explain why the internal energy and temperature of the gas increase during adiabatic compression. The work done compressing the gas increases its internal energy, illustrating a fundamental principle of energy transformation.