To solve the problem step by step, we need to determine the final volume of the gas after isothermal and irreversible expansion and calculate the work done during this process.
### Step 1: Identify the initial conditions
We are given:
- Initial volume (V1) = 2 dm³
- Initial pressure (P1) = 5 bar
- External pressure (P_ext) = 1 bar
### Step 2: Use the ideal gas law for isothermal processes
In an isothermal process for an ideal gas, the product of pressure and volume remains constant. Therefore, we can use the relationship:
\[ P_1 \times V_1 = P_2 \times V_2 \]
Where:
- \( P_2 \) is the final pressure (which will be equal to the external pressure during irreversible expansion, so \( P_2 = P_{ext} = 1 \) bar).
- \( V_2 \) is the final volume we need to find.
### Step 3: Substitute the known values into the equation
Substituting the known values into the equation:
\[ 5 \, \text{bar} \times 2 \, \text{dm}^3 = 1 \, \text{bar} \times V_2 \]
### Step 4: Solve for the final volume (V2)
Rearranging the equation to solve for \( V_2 \):
\[ V_2 = \frac{5 \, \text{bar} \times 2 \, \text{dm}^3}{1 \, \text{bar}} \]
Calculating this gives:
\[ V_2 = 10 \, \text{dm}^3 \]
### Step 5: Calculate the work done (W)
The work done during an expansion against a constant external pressure can be calculated using the formula:
\[ W = -P_{ext} \times \Delta V \]
Where:
- \( \Delta V = V_2 - V_1 \)
Calculating \( \Delta V \):
\[ \Delta V = 10 \, \text{dm}^3 - 2 \, \text{dm}^3 = 8 \, \text{dm}^3 \]
Now substituting into the work formula:
\[ W = -1 \, \text{bar} \times 8 \, \text{dm}^3 \]
### Step 6: Convert the work into joules
Since \( 1 \, \text{bar} = 100 \, \text{kPa} \) and \( 1 \, \text{dm}^3 = 0.001 \, \text{m}^3 \):
\[ W = -1 \, \text{bar} \times 8 \, \text{dm}^3 = -1 \times 100 \, \text{kPa} \times 8 \times 0.001 \, \text{m}^3 \]
Calculating this gives:
\[ W = -800 \, \text{J} \]
### Final Results
- Final Volume (V2) = 10 dm³
- Work Done (W) = -800 J
