Question

One gas of \(n_1\) mole of molecules at temperature \(T_1\), volume \(V_1\), and pressure \(P_1\), and another gas of \(n_2\) mole of molecules at temperature \(T_2\), volume \(V_2\), and pressure \(P_2\), are mixed resulting in pressure \(P\) and volume \(V\) of the mixture. The temperature of the mixture is ______.

• A. \((T_1 + T_2)/2\)
• B. \(T_1 T_2 PV/(T_2 P_1 V_1 + T_1 P_2 V_2)\)
• C. \((T_2 P_1 V_1 + T_1 P_2 V_2)/(T_1 T_2 PV)\)
• D. \(|T_1 - T_2|/2\)

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
When two ideal gases are mixed, the total number of moles in the mixture is the sum of the moles of the individual gases ($n = n_1 + n_2$). We use the ideal gas law $PV = nRT$ for each component and the final mixture.

Step 2: Key Formula or Approach:
1. For gas 1: $n_1 = \frac{P_1 V_1}{R T_1}$ 2. For gas 2: $n_2 = \frac{P_2 V_2}{R T_2}$ 3. For mixture: $n = \frac{PV}{RT}$

Step 3: Detailed Explanation:
1. Equate the total moles: \[ \frac{PV}{RT} = \frac{P_1 V_1}{R T_1} + \frac{P_2 V_2}{R T_2} \] 2. Cancel $R$ from both sides: \[ \frac{PV}{T} = \frac{P_1 V_1}{T_1} + \frac{P_2 V_2}{T_2} \] 3. Find a common denominator for the right side: \[ \frac{PV}{T} = \frac{P_1 V_1 T_2 + P_2 V_2 T_1}{T_1 T_2} \] 4. Rearrange to solve for $T$: \[ T = \frac{PV T_1 T_2}{P_1 V_1 T_2 + P_2 V_2 T_1} \]

Step 4: Final Answer:
The temperature of the mixture is \( \frac{T_1 T_2 PV}{T_2 P_1 V_1 + T_1 P_2 V_2} \).