1.5 mole of `N_(2)` at `77^(@)C` and at a pressure of 5 atmosphere is mixed with 0.5 mole of helium at `27^(@)C` and at a pressure of 2 atmosphere. The volume of the mixture is equal to the sum of their initial volume. What is the pressure of the mixture if the temperature of the mixture is `69^(@)C`.

Explanation

To solve the problem step by step, we will use the ideal gas law, which is given by the equation:

\[ PV = nRT \]

Where:
- \( P \) = pressure
- \( V \) = volume
- \( n \) = number of moles
- \( R \) = ideal gas constant
- \( T \) = temperature in Kelvin

### Step 1: Convert Temperatures to Kelvin
- For Nitrogen (\( N_2 \)):
\[ T_1 = 77^\circ C = 77 + 273 = 350 \, K \]
- For Helium (\( He \)):
\[ T_2 = 27^\circ C = 27 + 273 = 300 \, K \]
- For the mixture:
\[ T_f = 69^\circ C = 69 + 273 = 342 \, K \]

### Step 2: Calculate the Volume of Each Gas
Using the ideal gas law, we can find the volumes \( V_1 \) and \( V_2 \) for each gas.

1. **For Nitrogen (\( N_2 \))**:
\[ V_1 = \frac{n_1RT_1}{P_1} = \frac{1.5 \, \text{moles} \times R \times 350 \, K}{5 \, \text{atm}} = \frac{525R}{5} = 105R \]

2. **For Helium (\( He \))**:
\[ V_2 = \frac{n_2RT_2}{P_2} = \frac{0.5 \, \text{moles} \times R \times 300 \, K}{2 \, \text{atm}} = \frac{150R}{2} = 75R \]

### Step 3: Calculate the Total Volume of the Mixture
The total volume \( V_f \) of the mixture is the sum of the individual volumes:
\[ V_f = V_1 + V_2 = 105R + 75R = 180R \]

### Step 4: Calculate the Total Number of Moles in the Mixture
The total number of moles \( n_f \) in the mixture is:
\[ n_f = n_1 + n_2 = 1.5 + 0.5 = 2 \, \text{moles} \]

### Step 5: Calculate the Pressure of the Mixture
Now we can find the pressure \( P_f \) of the mixture using the ideal gas law:
\[ P_f = \frac{n_fRT_f}{V_f} \]
Substituting the known values:
\[ P_f = \frac{2 \, \text{moles} \times R \times 342 \, K}{180R} \]
The \( R \) cancels out:
\[ P_f = \frac{2 \times 342}{180} = \frac{684}{180} \approx 3.8 \, \text{atm} \]

### Final Answer
The pressure of the mixture is approximately **3.8 atm**.