Question

Find the final temperature of a mixture of two gases. If one gas has pressure \(P_1\), temperature \(T_1\), number of moles \(n_1\) and volume \(V_1\); and the second gas has pressure \(P_2\), temperature \(T_2\), number of moles \(n_2\) and volume \(V_2\). If the final pressure is \(P\) and final volume is \(V\), find the final temperature of the mixture.

• A. \( \displaystyle \frac{PV}{\frac{P_1V_1}{T_1}+\frac{P_2V_2}{T_2}} \)
• B. \( \displaystyle \frac{PV(T_1+T_2)}{P_1V_1+P_2V_2} \)
• C. \( \displaystyle \frac{(P_1V_1+P_2V_2)(T_1+T_2)}{PV} \)
• D. \( \displaystyle \frac{PV}{P_1V_1}T_1+\left(\frac{PV}{P_2V_2}\right)T_2 \)

Answer 1

The Correct Option is A

Concept:
For an ideal gas, \[ PV = nRT \] Thus, \[ n = \frac{PV}{RT} \] When two gases mix, the total number of moles is conserved. \[ n = n_1 + n_2 \] Step 1: Write number of moles for each gas. \[ n_1 = \frac{P_1V_1}{RT_1} \] \[ n_2 = \frac{P_2V_2}{RT_2} \] Step 2: Write total number of moles after mixing. \[ n = \frac{PV}{RT_f} \] Step 3: Use mole conservation. \[ \frac{PV}{RT_f} = \frac{P_1V_1}{RT_1} + \frac{P_2V_2}{RT_2} \] Cancel \(R\): \[ \frac{PV}{T_f} = \frac{P_1V_1}{T_1} + \frac{P_2V_2}{T_2} \] Step 4: Solve for \(T_f\). \[ T_f = \frac{PV}{\frac{P_1V_1}{T_1} + \frac{P_2V_2}{T_2}} \]