Problem 56 - University Physics with Modern Physics

Question

High-Altitude Research. A large research balloon containing \(2.00 \times 10^{3} \mathrm{m}^{3}\) of helium gas at 1.00 atm and a temperature of \(15.0^{\circ} \mathrm{C}\) rises rapidly from ground level to an altitude at which the atmospheric pressure is only 0.900 atm \((\text { Fig. } 19.33)\) . Assume the helium behaves like an ideal gas and the balloon's ascent is too rapid to permit much heat exchange with the surrounding air. (a) Calculate the volume of the gas at the higher altitude. (b) calculate the temperature of the gas at the higher altitude. (c) What is the change in internal energy of the helium as the balloon rises to the higher altitude?

Step-by-Step Solution

Answer

The final volume is approximately 2300 m³, the temperature decreases to about 264 K, and the change in internal energy is negative.

Step 1: Understand the Initial Conditions

We are given an initial volume of helium gas \( V_1 = 2.00 \times 10^{3} \ \mathrm{m}^3 \) at a pressure \( P_1 = 1.00 \ \mathrm{atm} \) and temperature \( T_1 = 15.0^{\circ} \mathrm{C} = 288 \ \mathrm{K} \). The gas undergoes an adiabatic process where \( Q = 0 \), meaning no heat is exchanged with the environment.

Step 2: Determine the Volume at Higher Altitude

Using the combined gas law, which states \( \frac{P_1 V_1}{T_1} = \frac{P_2 V_2}{T_2} \), and knowing the new pressure \( P_2 = 0.900 \ \mathrm{atm} \), we can solve for \( V_2 \). Since there is no heat exchange, let's start by finding \( T_2 \) first to effectively use the equation: \[ V_2 = \frac{P_1 V_1 T_2}{P_2 T_1} \].

Step 3: Calculate the Temperature at Higher Altitude

For an adiabatic process, the relationship is \( P_1 V_1^\gamma = P_2 V_2^\gamma \) and \( T V^{\gamma-1} = \text{constant} \). For monotonic gases like helium, \( \gamma = \frac{5}{3} \). Rearranging gives: \( T_2 = T_1 \left(\frac{P_2}{P_1}\right)^{\frac{\gamma-1}{\gamma}} \). Substitute known values to find \( T_2 \).

Step 4: Calculate the Final Volume

Substitute \( T_2 \) into the equation \[ V_2 = \frac{P_1 V_1 T_2}{P_2 T_1} \] to find the volume at the higher altitude. Calculate and solve for \( V_2 \).

Step 5: Calculate the Change in Internal Energy

The change in internal energy for an ideal gas is given by \( \Delta U = n C_V \Delta T \). Since \( n = \frac{P_1 V_1}{RT_1} = \frac{P_2 V_2}{RT_2} \), we need \( n \) and the change in temperature \( \Delta T = T_2 - T_1 \). Use \( C_V = \frac{3}{2}R \) for helium.

Key Concepts

Ideal Gas Law

The Ideal Gas Law is a fundamental principle in understanding how gases behave under various conditions of pressure, volume, and temperature. It is expressed as \( PV = nRT \), where \( P \) represents the pressure of the gas, \( V \) represents its volume, \( n \) is the amount of substance (in moles), \( R \) is the ideal gas constant, and \( T \) is the temperature in Kelvin. For a large research balloon containing helium gas, this law is crucial to calculate the changes in gas characteristics as the balloon rises.

This law assumes that the gas particles do not interact with each other and are in constant random motion. It also assumes that the size of the gas particles is negligible compared to the space between them. These assumptions are valid for an ideal gas, helping us to understand real gases under many conditions.

When applying this to the balloon scenario, the Ideal Gas Law is used to relate the initial and final states of the gas as it undergoes rapid ascent, experiencing changes in pressure and temperature.

Internal Energy

Internal energy refers to the total energy contained within a system, resulting from both the motion of the molecules and the interactions between them. In gases, this internal energy is mostly due to the kinetic energy of its particles. For ideal gases, the internal energy depends solely on temperature.

For ideal gases undergoing adiabatic processes (where no heat is exchanged with the environment), the change in internal energy can be calculated using the formula \( \Delta U = nC_V\Delta T \).

Here, \( C_V \) is the molar specific heat at constant volume, which for a monatomic gas like helium is \( \frac{3}{2}R \). This represents the amount of heat required to raise the temperature of the gas without changing its volume. The change in temperature \( \Delta T \) will affect the internal energy of the gas as it ascends to higher altitudes, given that the kinetic energy of the particles and thus the temperature will vary.

Thermodynamics

Thermodynamics is the study of heat, work, and energy, especially through the laws governing these processes in systems. It's crucial for understanding how the balloon's helium behaves as it ascends and the effects of environmental changes like pressure and temperature.

There are four laws of thermodynamics, but most scenarios involving gases like helium typically involve the first and second laws:

• The First Law of Thermodynamics, or the law of energy conservation, states that energy cannot be created or destroyed, only transformed. For the balloon, the energy transition happens through work done by the gas as it expands against the external pressure.
• The Second Law of Thermodynamics states that processes tend to move towards thermal equilibrium, where disorder or entropy increases. When the balloon rises rapidly, the adiabatic process implies little to no heat exchange, maintaining the entropy level constant.

These laws explain the balloon’s behavior throughout its ascent as an adiabatic process, characterized by changes in pressure, volume, and temperature, without heat being transferred.

Monatomic Gases

Monatomic gases, such as helium, consist of single atoms. They are different from diatomic or polyatomic gases, which consist of molecules rather than single atoms. Helium, being a monatomic gas, exhibits unique properties important for calculations relating to the behavior of gases.

In a monatomic gas, the internal energy is mainly the result of translational kinetic energy because there are no rotational or vibrational energies involved, unlike in more complex molecules. This simplifies calculations considerably. The specific heat capacities, \( C_P \) and \( C_V \), are given by \( \frac{5}{2}R \) and \( \frac{3}{2}R \), respectively, as monatomic gases have fewer degrees of freedom compared to diatomic or polyatomic gases.

Knowing these properties, we can use them to solve problems like determining how the gas's temperature changes during an adiabatic process when pressure changes. This is why helium’s behavior in the high-altitude research balloon is predictable and can be explained with the ideal gas and thermodynamics principles.