QUESTION:
A vessel of volume V = 30\rm \ l contains ideal gas at the temperature 0^0\rm \ C. After a portion of the gas has been let out, the pressure in the vessel decreased by \Delta P = 0.78\rm \ atm (the temperature remaining constant). Find the mass of the released gas. The gas density under the normal conditions \rho = 1.3\rm \ g/l.
To solve this problem, you must have an idea of the ideal gas law.
SOLUTION:
Let’s suppose:
- m_1 is the masses of the gas in the vessel before the gas is released,
- m_2 is the masses of the gas in the vessel after the gas is released,
- P_1, V_1, T_1 are the pressure, volume and temperature of the gas in the vessel before the gas is released, and
- P_2 is the pressure of the gas in the vessel after the gas is released.
Given:
- V_1 = V = 30\rm \ l
- T_1 = 0^0\rm \ C
- \Delta P = 0.78\rm \ atm
- \rho = 1.3\rm \ g/l
So, the mass of the gas released is
\begin{aligned}\Delta m &= m_1 –m_2\\\end{aligned}
Applying ideal gas law before the gas is released, we have
\begin{aligned}P_1V_1&=\dfrac{m_1}{M} RT_1 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (1)\\\end{aligned}
Applying ideal gas law after the gas is released, we have
\begin{aligned}P_2V_1 &= \dfrac{m_2}{M} RT_1 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (2)\\\end{aligned}
(It is given that the volume and temperature are same before and after the release of the gas from the vessel)
From equation (1) and (2)
\begin{aligned}(P_1-P_2)V_1 &= \dfrac{m_1-m_2}{M} RT_1\\&= \dfrac{\Delta m}{M} RT_1\\\Rightarrow \Delta m &= \dfrac{(P_1-P_2)V_1M}{RT_1}\\&= \dfrac{(\Delta PV_1M}{RT_1} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (3)\\\end{aligned}
At the normal condition, P_0 = 1\rm \ atm and T_0 = 273\rm \ K = 0^0 \ C = T_1. So, from ideal gas law
\begin{aligned}P_0V_0 &=\dfrac{m}{M} RT_0\\\Rightarrow P_0 &=\dfrac{m}{V_0M} RT_0\\&=\dfrac{\rho RT_0}{M}\\\Rightarrow \dfrac{M}{RT_0} &=\dfrac{\rho}{P_0}\\\end{aligned}
Therefore, from equation (3) we get
\Delta m = \dfrac{\rho V_1 \Delta P}{P_0} = \dfrac{(1.3\rm \ g/l) (30\rm \ l) (0.78\rm \ atm)}{ 1\rm \ atm }=30\rm \ g
