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The Ideal Gas Law is a crucial equation in chemistry and physics, describing the behavior of an ideal gas. It connects pressure (P), volume (V), temperature (T), and the number of moles (n) in a gas with the equation: \[ PV = nRT \] Here, \(R\) is the ideal gas constant, which is 8.314 J/(mol K) when working with SI units. This law helps us understand how gases respond to changes in their environment, such as temperature and pressure. In our exercise, we use a modified form of the Ideal Gas Law: \(\rho = \frac{PM}{RT}\). This form helps to find the density (\(\rho\)) of a gas, which is necessary for calculating the speed of sound. Understanding the Ideal Gas Law is essential when working with gases, as it lays the foundation for various calculations and predictions about gas behavior.

Molar mass is the weight of one mole of a substance, typically expressed in kg/mol or g/mol. It's a key factor in converting between the mass of a substance and the amount of substance in moles. In our exercise, the molar mass of carbon dioxide (COe\(_2\)) is given as 44 kg/mol, where each CO\(_2\) molecule is made up of one carbon atom and two oxygen atoms. Calculating molar mass involves adding the atomic masses of the individual elements found on the periodic table. For CO\(_2\): - Carbon (C) has an atomic mass of approximately 12 g/mol. - Oxygen (O) has an atomic mass of approximately 16 g/mol and since CO\(_2\) has two oxygen atoms, you multiply this by 2. Therefore, the molar mass calculation becomes: \[ M = 12 + (2 \times 16) = 44 \text{ g/mol} \] Converting to kg/mol (since 1 g = 0.001 kg), remains as 44 kg/mol, which is as stated in the problem.

Converting temperature measurements from Celsius to Kelvin is common in scientific calculations, as Kelvin is the standard unit of measurement in the SI (International System of Units). The relationship between Celsius and Kelvin is straightforward: \[ T(K) = T(^{\circ}C) + 273.15 \] This conversion is crucial because the Kelvin scale starts at absolute zero, the point where molecular motion stops, and ensures all scientific calculations remain positive. In this exercise, temperature conversion is applied as follows: - Given temperature in Celsius: 400°C - Converted to Kelvin: \[ T(K) = 400 + 273.15 = 673.15 \text{ K} \] By converting to Kelvin, we ensure that all mathematical operations are valid and in line with scientific standards.

Density (\(\rho\)) is the measure of mass per unit volume and is commonly expressed in kg/m³ for gases. Calculating the density of a gas is a crucial step in determining other properties, like the speed of sound in the gas, in this exercise. Using the Ideal Gas Law, we can express density as: \[ \rho = \frac{PM}{RT} \] Here, each symbol represents: - \(P\) is the pressure, which should be converted from atm to Pascals (1 atm = 101325 Pa). For our exercise: \[ P = 0.50 \times 101325 = 50662.5 \text{ Pa} \] - \(M\) is the molar mass. - \(R\) is the ideal gas constant. - \(T\) is the temperature in Kelvin.Inserting the values from the exercise into the formula gives the density that can be used for other calculations, ensuring precise results when examining gas behavior under specific conditions.