A closed tank is first evacuated and then connected to a 50 L cylinder containing compressed nitrogen gas. The gas pressure in the cylinder originally at 20.5 bar falls to 11.2 bar after it is connected to the evacuated tank. Calculate the volume of the tank.

Explanation

To solve the problem, we will use the ideal gas law and the concept of conservation of moles. Here’s a step-by-step breakdown of the solution:

### Step 1: Understand the Scenario
We have a closed tank that is initially evacuated (meaning it has no gas inside). Then, a 50 L cylinder containing nitrogen gas at a pressure of 20.5 bar is connected to the tank. After the connection, the pressure in the cylinder drops to 11.2 bar.

### Step 2: Identify the Variables
- **Initial pressure in the cylinder (P1)** = 20.5 bar
- **Final pressure in the cylinder (P2)** = 11.2 bar
- **Volume of the cylinder (V1)** = 50 L
- **Volume of the tank (V2)** = ? (this is what we need to find)

### Step 3: Use the Ideal Gas Law
The ideal gas law is given by the equation:
\[ PV = nRT \]
Where:
- \( P \) = pressure
- \( V \) = volume
- \( n \) = number of moles of gas
- \( R \) = ideal gas constant
- \( T \) = temperature (assumed constant)

### Step 4: Calculate the Number of Moles Initially
Initially, when the nitrogen gas is in the cylinder, the number of moles of nitrogen gas can be calculated using the initial pressure and volume:
\[ n = \frac{P_1 V_1}{RT} \]

### Step 5: Calculate the Number of Moles After Connection
After the tank is connected, the total volume becomes \( V1 + V2 \) and the pressure is now \( P2 \). The number of moles of nitrogen gas can be expressed as:
\[ n = \frac{P_2 (V_1 + V_2)}{RT} \]

### Step 6: Set Up the Equation
Since the number of moles of nitrogen gas remains constant before and after the connection, we can set the two equations equal to each other:
\[ \frac{P_1 V_1}{RT} = \frac{P_2 (V_1 + V_2)}{RT} \]

### Step 7: Cancel Out Common Terms
The \( RT \) terms on both sides can be canceled out:
\[ P_1 V_1 = P_2 (V_1 + V_2) \]

### Step 8: Substitute Known Values
Substituting the known values into the equation:
\[ 20.5 \times 50 = 11.2 \times (50 + V_2) \]

### Step 9: Solve for V2
Now, we can solve for \( V_2 \):
1. Calculate the left side:
\[ 20.5 \times 50 = 1025 \]
2. Expand the right side:
\[ 11.2 \times (50 + V_2) = 560 + 11.2 V_2 \]
3. Set the equation:
\[ 1025 = 560 + 11.2 V_2 \]
4. Rearranging gives:
\[ 1025 - 560 = 11.2 V_2 \]
\[ 465 = 11.2 V_2 \]
5. Finally, solve for \( V_2 \):
\[ V_2 = \frac{465}{11.2} \approx 41.5 \text{ L} \]

### Conclusion
The volume of the tank is approximately **41.5 L**.