Question

Consider a well-insulated horizontal rigid cylinder that is divided into two compartments by a piston that is free to move but does not allow either gas to leak into the other side. Initially, one side of the piston contains \(1 \mathrm{m}^{3}\) of \(\mathrm{N}_{2}\) gas at \(500 \mathrm{kPa}\) and \(120^{\circ} \mathrm{C}\) while the other side contains \(1 \mathrm{m}^{3}\) of He gas at \(500 \mathrm{kPa}\) and \(40^{\circ} \mathrm{C}\). Now thermal equilibrium is established in the cylinder as a result of heat transfer through the piston. Using constant specific heats at room temperature, determine the final equilibrium temperature in the cylinder. What would your answer be if the piston were not free to move?

Short answer

Answer: To find the final equilibrium temperature, follow these steps: 1. Determine the initial conditions for both nitrogen and helium gas in terms of volume, pressure, and temperature. 2. Calculate the number of moles of each gas using the ideal gas law. 3. Calculate the initial internal energies for each gas using their respective moles, specific heat capacities, and temperatures. 4. Apply the principle of conservation of energy to find the final temperature for each gas after heat transfer takes place. 5. If the piston is not free to move, calculate the final temperature for each gas by applying the relation for an ideal gas undergoing an adiabatic process.

Step-by-step solution

Initial Conditions

Given the initial conditions for nitrogen (N₂) gas: Volume (V₁) = 1 m³ Pressure (P₁) = 500 kPa Temperature (T₁) = 120 °C + 273.15K = 393.15K And for helium (He) gas: Volume (V₂) = 1 m³ Pressure (P₂) = 500 kPa Temperature (T₂) = 40 °C + 273.15K = 313.15K Step 2: Calculate the number of moles using the ideal gas law.

Moles of Gases

Use the ideal gas law to find the number of moles (n) of each gas: PV = nRT For N₂ gas: n₁ = P₁V₁ / (R₁T₁) For He gas: n₂ = P₂V₂ / (R₂T₂) Use the specific gas constant, R₁= 296.8 J/kg.K for N₂ gas and R₂= 2077 J/kg.K for He gas. Step 3: Calculate the initial internal energies.

Internal Energies

Using the equation for internal energy (U) of an ideal gas: U₁ = n₁Cv₁T₁ U₂ = n₂Cv₂T₂ Use constant specific heat values for nitrogen: Cv₁ = 0.743 kJ/kg.K and for helium: Cv₂ = 3.12 kJ/kg.K. Step 4: Apply conservation of energy.

Conservation of Energy

The total internal energy is constant before and after thermal equilibrium: U_total_initial = U₁ + U₂ = n₁Cv₁T₁ + n₂Cv₂T₂ And after thermal equilibrium: U_total_final = U₁' + U₂' = n₁Cv₁T + n₂Cv₂T Solve for the final temperature (T). Step 5: Calculate the final temperature if the piston is not free to move.

Fixed Piston Scenario

If the piston is not free to move, the initial internal energy will not change since the process would be adiabatic. Calculate the final temperature T₃ if the gas undergoes an adiabatic process and use the relation for an ideal gas: Cv(T₃-T₂) = -n₁Cv₁ ln(T₃/T₁) Solve for T₃.