A fixed mass of an ideal gas of volume 50 litre measured at 2 atm and `0^(@)C`. At the same temperature but at 5 atm its volume will be

Options

A. 20 litres
B. 40 litres
C. 60 litres
D. 80 litres

Answer

To solve the problem of finding the volume of an ideal gas at a different pressure while keeping the temperature constant, we can use Boyle's Law. Here’s a step-by-step solution:

### Step 1: Understand Boyle's Law
Boyle's Law states that for a given mass of an ideal gas at constant temperature, the pressure (P) of the gas is inversely proportional to its volume (V). Mathematically, this can be expressed as:
\[ P_1 V_1 = P_2 V_2 \]
where:
- \( P_1 \) = initial pressure
- \( V_1 \) = initial volume
- \( P_2 \) = final pressure
- \( V_2 \) = final volume

### Step 2: Identify Given Values
From the problem, we have:
- Initial pressure, \( P_1 = 2 \, \text{atm} \)
- Initial volume, \( V_1 = 50 \, \text{liters} \)
- Final pressure, \( P_2 = 5 \, \text{atm} \)
- Final volume, \( V_2 \) = ? (this is what we need to find)

### Step 3: Substitute Values into Boyle's Law
Now, we can substitute the known values into the equation:
\[ 2 \, \text{atm} \times 50 \, \text{liters} = 5 \, \text{atm} \times V_2 \]

### Step 4: Solve for \( V_2 \)
Rearranging the equation to solve for \( V_2 \):
\[ V_2 = \frac{2 \, \text{atm} \times 50 \, \text{liters}}{5 \, \text{atm}} \]

### Step 5: Calculate \( V_2 \)
Now, performing the calculation:
\[ V_2 = \frac{100 \, \text{atm} \cdot \text{liters}}{5 \, \text{atm}} = 20 \, \text{liters} \]

### Conclusion
Thus, the volume of the gas at 5 atm and at the same temperature is:
\[ V_2 = 20 \, \text{liters} \]

### Final Answer
The volume will be **20 liters**.

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