Question

An ideal gas is placed in a container at \( (P_1, V_1, T_1) \) and another ideal gas is placed in a different container at \( (P_2, V_2, T_2) \) are mixed at final pressure of \( P \) and final volume of \( V \). Calculate the final temperature.

• A. \( \frac{T_1 T_2}{P_1 V_1 T_2 + P_2 V_2 T_1} \cdot \frac{1}{PV} \)
• B. \( \frac{T_1 T_2}{P_1 V_1 T_2 + P_2 V_2 T_1} \cdot PV \)
• C. \( \frac{P_1 V_1 T_2 + P_2 V_2 T_1}{T_1 T_2} \cdot PV \)
• D. \( \frac{P_1 V_1 + P_2 V_2}{T_1 T_2} \cdot \frac{1}{PV} \)

Answer 1

The Correct Option is B

textbf{Step 1: Use Ideal Gas Law.}
The ideal gas law states: \[ P V = n R T \] where \( P \) is the pressure, \( V \) is the volume, \( n \) is the number of moles, and \( T \) is the temperature. Step 2: Combine the Ideal Gas Laws for both gases.
Let the number of moles of gas in the first container be \( n_1 \) and in the second container be \( n_2 \). The ideal gas law for each gas before mixing is: \[ P_1 V_1 = n_1 R T_1 \quad \text{and} \quad P_2 V_2 = n_2 R T_2 \] After mixing, the total pressure is \( P \) and the total volume is \( V \). Using the ideal gas law for the final state, we have: \[ P V = (n_1 + n_2) R T_f \] where \( T_f \) is the final temperature. Step 3: Solve for the final temperature.
From the equations above, we get: \[ n_1 = \frac{P_1 V_1}{R T_1} \quad \text{and} \quad n_2 = \frac{P_2 V_2}{R T_2} \] Substitute these values into the equation for the final temperature: \[ P V = \left( \frac{P_1 V_1}{R T_1} + \frac{P_2 V_2}{R T_2} \right) R T_f \] Simplifying this gives: \[ T_f = \frac{T_1 T_2}{P_1 V_1 T_2 + P_2 V_2 T_1} \cdot PV \] \[ \boxed{\frac{T_1 T_2}{P_1 V_1 T_2 + P_2 V_2 T_1} \cdot PV} \]