FIGURE CP21.70 shows two insulated compartments separated by a thin wall. The left side contains 0.060 mol of helium at an initial temperature of 600 K and the right side contains 0.030 mol of helium at an initial temperature of 300 K. The compartment on the right is attached to a vertical cylinder, above which the air pressure is 1.0 atm. A 10-cm-diameter, 2.0 kg piston can slide without friction up and down the cylinder. Neither the cylinder diameter nor the volumes of the compartments are known. How much heat is transferred from the left side to the right side?

Verified step by step guidance

1. Step 1: Identify the key principles involved in the problem. This problem involves the transfer of heat between two systems containing helium gas. The gases are ideal, and the heat transfer will occur until thermal equilibrium is reached. The piston on the right side is affected by external pressure and its own weight, which will influence the final state of the system.
2. Step 2: Write the equation for thermal equilibrium. When the two systems reach equilibrium, their final temperatures will be the same. Use the principle of conservation of energy: the heat lost by the left compartment (System 1) will equal the heat gained by the right compartment (System 2). The equation is: \( Q_1 = -Q_2 \).
3. Step 3: Use the formula for heat transfer for an ideal gas. The heat transferred in each system can be calculated using \( Q = n C_v \Delta T \), where \( n \) is the number of moles, \( C_v \) is the molar specific heat at constant volume for helium (\( C_v = \frac{3}{2}R \)), and \( \Delta T \) is the change in temperature. Write expressions for \( Q_1 \) and \( Q_2 \) in terms of \( T_f \), the final temperature.
4. Step 4: Account for the piston dynamics. The piston on the right side is affected by the external pressure (1 atm) and its own weight. The pressure inside the right compartment must balance these forces. Use the ideal gas law \( PV = nRT \) to relate the pressure, volume, and temperature of the gas in the right compartment. The piston’s equilibrium position will determine the final volume of the right compartment.
5. Step 5: Solve for the final temperature \( T_f \) and calculate the heat transferred. Combine the equations for heat transfer and the ideal gas law to solve for \( T_f \). Once \( T_f \) is determined, substitute it back into the heat transfer equations to find the amount of heat transferred from System 1 to System 2.

Key Concepts

Thermodynamics

Thermodynamics is the branch of physics that deals with heat, work, temperature, and the laws governing energy transfer. It is essential for understanding how energy moves between systems, particularly in processes involving heat transfer, such as the one described in the question. The first and second laws of thermodynamics provide the framework for analyzing energy conservation and entropy changes in the system.

Ideal Gas Law

The Ideal Gas Law relates the pressure, volume, temperature, and number of moles of an ideal gas through the equation PV = nRT. In this scenario, it helps to determine the behavior of helium gas in both compartments as they exchange heat and work is done by the piston. Understanding this relationship is crucial for calculating changes in temperature and pressure during the heat transfer process.

Heat Transfer

Heat transfer refers to the movement of thermal energy from one object or system to another due to a temperature difference. In this case, heat will flow from the hotter compartment (600 K) to the cooler one (300 K) until thermal equilibrium is reached. The amount of heat transferred can be calculated using specific heat capacities and the change in temperature, which is vital for solving the problem presented.