A rigid vessel of volume ` 0.50m^(3)` containing `H_(2) ` at `20.5^(@)C` and a pressure of `611 xx 10^(3)` Pa is connected to a second rigid vessel of volume 0.75`m^(3)` containing Ar at `31.2^(@)C` at a pressure of `433xx10^(3)` Pa. A valve separating the two vessels is opened and both are cooled to a temperature of `14.5^(@)C`. What is the final pressure in the vessels?

Options

A. `2xx10^(5)`
B. `3.22xx10^(5)" Pa"`
C. 4840 pa
D. `4.84xx10^(5)" Pa"`

Explanation

To solve the problem step by step, we will use the ideal gas law, which states that \( PV = nRT \).

### Step 1: Convert temperatures to Kelvin
We need to convert the temperatures from Celsius to Kelvin for both vessels.

- For \( H_2 \):
\[ T_1 = 20.5^\circ C + 273.15 = 293.65 \, K \]

- For \( Ar \):
\[ T_2 = 31.2^\circ C + 273.15 = 304.35 \, K \]

- Final temperature after cooling:
\[ T_f = 14.5^\circ C + 273.15 = 287.65 \, K \]

### Step 2: Calculate the number of moles in each vessel
Using the ideal gas law, we can calculate the number of moles \( n \) in each vessel before the valve is opened.

- For \( H_2 \):
\[ n_1 = \frac{P_1 V_1}{RT_1} = \frac{611 \times 10^3 \, \text{Pa} \times 0.50 \, \text{m}^3}{R \times 293.65 \, K} \]

- For \( Ar \):
\[ n_2 = \frac{P_2 V_2}{RT_2} = \frac{433 \times 10^3 \, \text{Pa} \times 0.75 \, \text{m}^3}{R \times 304.35 \, K} \]

### Step 3: Combine the moles after the valve is opened
The total number of moles after the valve is opened is:
\[ n_{total} = n_1 + n_2 \]

### Step 4: Calculate the total volume
The total volume \( V_{total} \) after the valve is opened is the sum of the volumes of both vessels:
\[ V_{total} = V_1 + V_2 = 0.50 \, \text{m}^3 + 0.75 \, \text{m}^3 = 1.25 \, \text{m}^3 \]

### Step 5: Calculate the final pressure using the ideal gas law
Using the ideal gas law for the total system at the final temperature:
\[ P_f = \frac{n_{total} R T_f}{V_{total}} \]

### Step 6: Substitute the values and solve
Substituting the values we calculated earlier into the equation for \( P_f \):
\[ P_f = \frac{(n_1 + n_2) R T_f}{V_{total}} \]

After calculating \( n_1 \) and \( n_2 \) and substituting them into the equation, we can find \( P_f \).

### Final Result
After performing all calculations, we find:
\[ P_f \approx 484,000 \, \text{Pa} \, \text{or} \, 4.84 \times 10^5 \, \text{Pa} \]