A cylinder contained 10 kg of gas at pressure 10⁷ N /m² . The quantity of gas taken out of cylinder if final pressure is 2.5x10⁶ N /m is (assume the temperature of gas is constant)

The correct option is 7.5 kg.

Step-by-Step Explanation:

We are given the following data:
Initial mass of the gas, $m_1=10\text{kg}$
Initial pressure of the gas, $P_1={10}^7{\text{N/m}}^2$
Final pressure of the gas, $P_2=2.5\times {10}^6{\text{N/m}}^2$
Temperature of the gas ($T$) and volume of the cylinder ($V$) remain constant.

According to the ideal gas equation:
$PV=nRT$
Since the number of moles $n=\frac{m}{M}$ (where $m$ is the mass of the gas and $M$ is its molar mass), we can write:
$PV=\frac{m}{M}RT$

Since volume $V$, molar mass $M$, gas constant $R$, and temperature $T$ are all constant, pressure $P$ is directly proportional to the mass of the gas $m$ remaining in the cylinder:
$P\propto m$

Therefore, we can write the ratio of final state to initial state as:
$\frac{P_2}{P_1}=\frac{m_2}{m_1}$

Substituting the given values into the equation to find the final mass of the gas ($m_2$) remaining in the cylinder:
$\frac{2.5\times {10}^6}{{10}^7}=\frac{m_2}{10}$

Simplifying the left-hand side:
$\frac{2.5\times {10}^6}{10\times {10}^6}=\frac{m_2}{10}$
$0.25=\frac{m_2}{10}$
$m_2=0.25\times 10=2.5\text{kg}$

The final mass of the gas remaining in the cylinder is $2.5\text{kg}$.

The quantity of gas taken out of the cylinder ($\Delta m$) is the difference between the initial mass and the remaining mass:
$\Delta m=m_1-m_2$
$\Delta m=10\text{kg}-2.5\text{kg}=7.5\text{kg}$

Thus, the quantity of gas taken out of the cylinder is 7.5 kg.