Two vessels A and B thermally insulated, contain an ideal monoatomic gas. A small tube with valve connects A and B. Initialy A has 2 litres of gas at 300 K and 2*10⁵Pa pressure,while the vessel B has 4litres of gas at 350 K and 4*10⁵Pa pressure. The valve is now opened and the system attained equillibrium in pressure and temperature. Calculate the equillibrium pressure and temperature of the valve.

To solve the problem of finding the equilibrium pressure and temperature when the valve between two thermally insulated vessels is opened, we need to apply the principles of thermodynamics and the ideal gas law. Let's break this down step by step.

Understanding the Initial Conditions

We have two vessels, A and B, with the following properties:

• Vessel A: Volume = 2 L, Temperature = 300 K, Pressure = 2 × 10⁵ Pa
• Vessel B: Volume = 4 L, Temperature = 350 K, Pressure = 4 × 10⁵ Pa

Using the Ideal Gas Law

The ideal gas law states that:

P V = n R T

Where:

• P = pressure
• V = volume
• n = number of moles of gas
• R = ideal gas constant (8.314 J/(mol·K))
• T = temperature in Kelvin

Calculating the Number of Moles in Each Vessel

First, we need to calculate the number of moles of gas in each vessel using the ideal gas law.

For Vessel A:

Rearranging the ideal gas law gives us:

n_A = P_A V_A / (R T_A)

Substituting the values:

n_A = (2 × 10⁵ Pa) × (2 × 10^-3 m³) / (8.314 J/(mol·K) × 300 K)

Calculating this gives:

n_A ≈ 0.16 moles

For Vessel B:

Using the same formula:

n_B = P_B V_B / (R T_B)

Substituting the values:

n_B = (4 × 10⁵ Pa) × (4 × 10^-3 m³) / (8.314 J/(mol·K) × 350 K)

Calculating this gives:

n_B ≈ 0.55 moles

Finding the Total Number of Moles

The total number of moles in the system after the valve is opened is:

n_total = n_A + n_B ≈ 0.16 + 0.55 = 0.71 moles

Calculating the Equilibrium Pressure

At equilibrium, the pressure will be the same in both vessels. The total volume of the system is:

V_total = V_A + V_B = 2 L + 4 L = 6 L = 6 × 10^-3 m³

Using the ideal gas law again for the total system at equilibrium:

P_eq = n_total R T_eq / V_total

Finding the Equilibrium Temperature

To find the equilibrium temperature, we can use the concept of energy conservation. The total internal energy of the system must remain constant since it is thermally insulated. The internal energy for a monoatomic ideal gas is given by:

U = (3/2) n R T

Calculating the initial internal energy for both vessels:

For Vessel A:

U_A = (3/2) × n_A × R × T_A

U_A = (3/2) × 0.16 × 8.314 × 300 ≈ 62.4 J

For Vessel B:

U_B = (3/2) × n_B × R × T_B

U_B = (3/2) × 0.55 × 8.314 × 350 ≈ 69.5 J

Total Internal Energy

U_total = U_A + U_B ≈ 62.4 + 69.5 ≈ 131.9 J

Setting Up the Equation for Equilibrium

At equilibrium, the total internal energy can also be expressed as:

U_total = (3/2) n_total R T_eq

Setting the two expressions for total internal energy equal gives:

131.9 = (3/2) × 0.71 × 8.314 × T_eq

Solving for T_eq:

T_eq ≈ 131.9 / ((3/2) × 0.71 × 8.314) ≈ 131.9 / 2.95 ≈ 44.7 K

Final Equilibrium Values

Now, substituting T_eq back into the pressure equation:

P_eq = n_total</