A cylinder contained `10kg`of gas at pressure `10^(7) N / m^(2)`. The quantity of gas taken out of cylinder if final pressure is `2.5 xx 10^(6) N//m^(2)` is (Assume temperature of gas is constant)

Options

A. `9.5 ` kg
B. `7.5` kg
C. `14.2` kg
D. zero

Explanation

To solve the problem step by step, we will use the relationship between the number of moles of gas and pressure, given that the temperature and volume are constant.

### Step 1: Understand the relationship between moles and pressure
According to the ideal gas law, if the temperature (T) and volume (V) are constant, the number of moles (n) of gas is directly proportional to the pressure (P). This can be expressed as:
\[ n \propto P \]

### Step 2: Set up the initial and final conditions
Let:
- \( n_1 \) = initial number of moles of gas
- \( P_1 = 10^7 \, \text{N/m}^2 \) (initial pressure)
- \( P_2 = 2.5 \times 10^6 \, \text{N/m}^2 \) (final pressure)

### Step 3: Use the proportionality to find the final number of moles
From the proportionality, we can write:
\[ \frac{n_1}{n_2} = \frac{P_1}{P_2} \]
Rearranging gives:
\[ n_2 = n_1 \cdot \frac{P_2}{P_1} \]

### Step 4: Substitute the values of pressures
Substituting the values of \( P_1 \) and \( P_2 \):
\[ n_2 = n_1 \cdot \frac{2.5 \times 10^6}{10^7} = n_1 \cdot \frac{2.5}{10} = n_1 \cdot 0.25 \]

### Step 5: Calculate the change in number of moles
The quantity of gas taken out (change in moles, \( \Delta n \)) is given by:
\[ \Delta n = n_1 - n_2 = n_1 - (0.25 n_1) = n_1 (1 - 0.25) = n_1 \cdot 0.75 \]

### Step 6: Find the initial number of moles
We know the mass of the gas is \( 10 \, \text{kg} \). To find the number of moles \( n_1 \), we need the molar mass \( m \) of the gas:
\[ n_1 = \frac{\text{mass}}{\text{molar mass}} = \frac{10 \, \text{kg}}{m} \]

### Step 7: Substitute to find the quantity of gas taken out
Now substituting \( n_1 \) into the equation for \( \Delta n \):
\[ \Delta n = 0.75 \cdot \frac{10}{m} \]

### Step 8: Calculate the mass of gas taken out
The mass of gas taken out \( \Delta m \) can be calculated as:
\[ \Delta m = \Delta n \cdot m = 0.75 \cdot \frac{10}{m} \cdot m = 0.75 \cdot 10 = 7.5 \, \text{kg} \]

### Conclusion
The quantity of gas taken out of the cylinder is \( 7.5 \, \text{kg} \).