Describe how you would measure, by either chemical or physical means, the partial pressures of a mixture of gases of the following composition: (a) \(\mathrm{CO}_{2}\) and \(\mathrm{H}_{2,}(\mathrm{~b}) \mathrm{He}\) and \(\mathrm{N}_{2}\)

When exploring the properties of gases, the ideal gas law provides crucial insight. It's an equation that relates the pressure, volume, and temperature of a gas to the amount of substance present, typically measured in moles. Mathematically, the formula is expressed as: \[ PV = nRT \] where \( P \) represents the pressure, \( V \) is the volume, \( n \) indicates the number of moles, \( R \) is the ideal gas constant, and \( T \) corresponds to the temperature in Kelvin.

This relationship is particularly useful when considering the measurement of partial pressures in a gas mixture. Since each gas in a mixture behaves independently, the partial pressure of a single gas can be calculated assuming ideal behavior. Understanding this concept allows students to better appreciate how measurements like the one described in the original exercise can be deduced.

A gas analyser is a versatile instrument employed to determine the composition of gas mixtures. Often used in environmental monitoring and industrial applications, gas analysers can detect and quantify various gases from complex mixtures. They function based on different principles such as infrared, electrochemical, or thermal conductivity.

For instance, when examining a mixture of carbon dioxide and hydrogen, a gas analyser can identify the concentration of each. By knowing the concentration, the partial pressures are deduced using the ideal gas law. This linkage between the gas analyser's readings and the theoretical calculations makes such instrumentation indispensable in the practical assessment of gas compositions.

The mass spectrometer is an advanced analytical tool that distinguishes and measures the mass-to-charge ratio of ions. It operates by ionizing chemical species and separating the ions based on their mass-to-charge ratio using an electromagnetic field. This ability makes the mass spectrometer exceptionally suited for analyzing mixtures of gases like helium and nitrogen.

By ionizing the gas molecules and then separating them, the device provides precise measurements of the partial pressures of individual gases. The results from a mass spectrometer can verify the mole fraction and presence of each gas, making it a valuable apparatus for scientists and engineers in fields such as chemistry and environmental science.

Pressure sensors are critical components in measuring the physical properties of gases. They convert the physical quantity of pressure into an electrical signal that can be easily read and interpreted. These sensors come in many forms, including piezoelectric, capacitive, and strain-gauge based.

For mixed gas compositions, total pressure is initially obtained via a pressure sensor. Subsequently, the specific contributions of individual gases are extracted by applying our knowledge of their proportions, which are described by the mole fraction. This physical method of measuring partial pressures is a direct application of the sensations captured by the pressure sensors, highlighting the practical side of gas-related principles.

Mole fraction is a dimensionless quantity that represents the ratio of the number of moles of a particular component to the total number of moles in a mixture. It's essential in calculating partial pressures as it helps to identify the contribution of each gas within a mixture.

Mathematically, the mole fraction \( x_i \) of a gas \( i \) is given by: \[ x_i = \frac{n_i}{n_{total}} \] where \( n_i \) is the number of moles of gas \( i \), and \( n_{total} \) is the total number of moles of all gases in the mixture. When it comes to determining the partial pressure of gas \( i \), the mole fraction gets multiplied by the total pressure of the mixture: \[ P_i = x_i \times P_{total} \] This straightforward method is a potent tool for simulating and predicting the behavior of gas mixtures, catering to the understanding of phenomena ranging from simple everyday experiences to complex industrial processes.