Gaseous State
3.0 Ideal gas equation
3.0 Ideal gas equation
Gas which follows above gas laws at every temperature and pressure is considered to be an ideal gas. By combining above gas laws, a single equation is obtained which is known as ideal gas equation. Also all the gas laws can be obtained from ideal gas equation.
From boyle's law, $$V \propto 1/P$$ From charle's law, $$V \propto T$$ From avogadro's law, $$V \propto n$$
Combining all the above three equations, we get $$\begin{equation} \begin{aligned} V \propto Tn/P \\ V = RTn/P \\ PV = nRT \ ( with \ R \ as\ proportionality \ constant)\\\end{aligned} \end{equation} $$
$R$ is also known as 'Universal gas constant'. The value of $R$ in different unit system is different. $$\begin{equation} \begin{aligned} R = 8.314\ N - m{K^{ - 1}}mo{l^{ - 1}} \\ R = 0.00821\ atm - litre\ {K^{ - 1}}mo{l^{ - 1}} \\ R = 1.98\ cal\ {K^{ - 1}}mo{l^{ - 1}} \\\end{aligned} \end{equation} $$
If gas $1$ and $2$ having same number of moles and having $({P_{1,}}{V_1},{T_1})$ and $({P_{2,}}{V_2},{T_2})$ as pressure, volume and temperature respectively then the six variables can be related as,$${P_1}{V_1}/{T_1} = {P_2}{V_2}/{T_2}$$The above equation is known as combined gas law .
We know that, $n = m/M$. Substituting in ideal gas equation, we get $$\begin{equation} \begin{aligned} PV = (m/M)RT \\ PM = (m/V)RT \\ PM = \rho RT \\\end{aligned} \end{equation} $$ In this way, ideal gas equation can be expressed in terms of density of a gas.
Question 2. A vessel of $120$ mL capacity contains a certain amount of gas at ${35^ \circ }C$ and $1.2$ bar pressure. The gas is transferred to another vessel of volume $180$ mL at ${105^ \circ }C$. What would be its pressure?
Solution: From combined gas law equation,$$\begin{equation} \begin{aligned} {P_1}{V_1}/{T_1} = {P_2}{V_2}/{T_2} \\ 1.2 \times 120/308.15 = {P_2} \times 180/378.15 \\ {P_2} = 0.98bar \\\end{aligned} \end{equation} $$
