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 gas samples are mixed, the total number of moles is conserved. We can use the ideal gas equation (\(PV = nRT\)) to relate the final macroscopic parameters of the mixture to the individual components.
Step 2: Key Formula or Approach:
Total moles in the mixture: \(n_{\text{total}} = n_1 + n_2\).
For the first gas: \(n_1 = \frac{P_1 V_1}{R T_1}\).
For the second gas: \(n_2 = \frac{P_2 V_2}{R T_2}\).
For the final mixture: \(n_{\text{total}} = \frac{P V}{R T}\).
Step 3: Detailed Explanation:
Substitute the expressions for \(n_1, n_2,\) and \(n_{\text{total}}\) into the conservation of moles equation:
\(\frac{P V}{R T} = \frac{P_1 V_1}{R T_1} + \frac{P_2 V_2}{R T_2}\)
Cancel the universal gas constant \(R\) from both sides:
\(\frac{P V}{T} = \frac{P_1 V_1}{T_1} + \frac{P_2 V_2}{T_2}\)
Find a common denominator for the right hand side:
\(\frac{P V}{T} = \frac{P_1 V_1 T_2 + P_2 V_2 T_1}{T_1 T_2}\)
Now, invert both sides to solve for the final temperature \(T\):
\(\frac{T}{P V} = \frac{T_1 T_2}{P_1 V_1 T_2 + P_2 V_2 T_1}\)
\(T = \frac{P V T_1 T_2}{P_1 V_1 T_2 + P_2 V_2 T_1}\)
Step 4: Final Answer:
The temperature of the mixture is \(\frac{T_1 T_2 PV}{T_2 P_1 V_1 + T_1 P_2 V_2}\).