Derive the ideal gas equation on the basis of laws of Boyle, Charle and Avogadro. Calculate the volume occupied by an ideal gas at STP, if \(0.25 \ell\) of the gas is present at a pressure of \(700 \mathrm{~mm}\) of \(\mathrm{Hg}\) and \(273^{\circ} \mathrm{C}\).

To derive the ideal gas equation from Boyle's Law, Charles' Law, and Avogadro's Law, we will first briefly introduce these laws: - Boyle's Law: It states that the pressure of a given amount of gas is inversely proportional to its volume if the temperature remains constant, or \(PV=k_1\) where \(k_1\) is a constant. - Charles' Law: It states that the volume of a given amount of gas is directly proportional to its temperature if the pressure remains constant, or \(V/T=k_2\) where \(k_2\) is a constant. - Avogadro's Law: It states that the volume of a gas is directly proportional to the number of moles of the gas if pressure and temperature remain constant, or \(V/n=k_3\) where \(k_3\) is a constant. Now, we can combine these laws to derive the ideal gas equation. Let's represent the proportionalities with constants \(k_1, k_2, k_3\). From Boyle's Law, \(PV=k_1\) (1) From Charles' Law, \(V/T=k_2\) (2) From Avogadro's Law, \(V/n=k_3\) (3) Now, let's combine equations (1), (2), and (3) to form a single equation: \(PV=k_1(\frac{T}{k_2})(\frac{k_3}{n})\) Now, let's define the gas constant \(R=\frac{k_1 k_3}{k_2}\), then the above equation becomes: \(PV=nRT\) This is the ideal gas equation we were looking for.

Now, we will use the ideal gas equation to calculate the volume occupied by the given gas at STP. The given conditions are: - \(0.25 \ell\) of gas - Pressure of \(700 \mathrm{~mm}\) of \(\mathrm{Hg}\) - Temperature of \(273^{\circ} \mathrm{C}\) First, we need to convert the pressure and temperature to SI units: - Pressure: \(700 \mathrm{~mm} \mathrm{Hg} = 700 \times \frac{101.325 \mathrm{kPa}}{760 \mathrm{~mm} \mathrm{Hg}} = 92.664 \mathrm{kPa}\) - Temperature: \(273^{\circ} \mathrm{C} = 273+273 = 546 \mathrm{K}\) Now, let's use the ideal gas equation to find the number of moles of the gas: \(PV=nRT \Rightarrow n=\frac{PV}{RT}\) \(n = \frac{92.664 \mathrm{kPa} \times 0.25 \ell}{8.314 \mathrm{J\,mol^{-1}\,K^{-1}} \times 546 \mathrm{K}} = 0.00628 \mathrm{mol}\) Now, we will use the ideal gas equation again, but this time at STP conditions (pressure of \(101.325 \mathrm{kPa}\) and temperature of \(273 \mathrm{K}\)): \(PV=nRT \Rightarrow V=\frac{nRT}{P}\) \(V = \frac{0.00628 \mathrm{mol} \times 8.314 \mathrm{J\,mol^{-1}\,K^{-1}} \times 273 \mathrm{K}}{101.325 \mathrm{kPa}} = 0.141 \ell\) So, the volume occupied by the ideal gas at STP is approximately \(0.141 \ell\).