(a) Calculate the density of \(\mathrm{NO}_{2}\) gas at \(0.970 \mathrm{~atm}\) and \(35^{\circ} \mathrm{C}\). (b) Calculate the molar mass of a gas if \(2.50 \mathrm{~g}\) occupies \(0.875 \mathrm{~L}\) at 685 torr and \(35^{\circ} \mathrm{C}\).

We need to make sure the temperature is measured in Kelvin for the gas law calculations. To convert Celsius to Kelvin, add 273.15 to the given temperature: \(T_K = T_C + 273.15\) \(T_K = 35 + 273.15 = 308.15 \mathrm{K}\)

To calculate the molar mass of NO₂, we add the molar masses of one nitrogen atom and two oxygen atoms: Molar mass of N = 14.01 g/mol Molar mass of O = 16.00 g/mol Molar mass of NO₂ = 14.01 + 2(16.00) = 46.01 g/mol

The Ideal Gas Law is given by \(PV=nRT\), where P is the pressure, V is the volume, n is the number of moles, R is the ideal gas constant (0.0821 L atm/mol K), and T is the temperature. We'll modify the equation to solve for density (ρ), which is given by \(\rho =\frac{mass}{volume}\): \(\rho = \frac{n \times Molar\: mass}{V}\) Replace n/V from Ideal Gas Law equation, \(\frac{n}{V} = \frac{P}{RT}\) So, \(\rho = \frac{P \times Molar\:mass}{R \times T}\) Now plug in the values: \(\rho = \frac{0.970 \mathrm{atm} \times 46.01 \mathrm{g/mol}}{0.0821 \mathrm{L \times atm/mol \times K} \times 308.15 \mathrm{K}}\) After calculating, we get the density: \(\rho \approx 1.77 \mathrm{g/L}\) b) Calculate the molar mass of a gas if 2.50 g occupies 0.875 L at 685 torr and 35°C.

We need to make sure the pressure is measured in atm and temperature is measured in Kelvin for the gas law calculations: Pressure in atm = \(685 \mathrm{torr} \times \frac{1 \mathrm{atm}}{760 \mathrm{torr}} \approx 0.901 \mathrm{atm}\) Temperature in Kelvin, \(T_K = T_C + 273.15\) \(T_K = 35 + 273.15 = 308.15 \mathrm{K}\)

Again, use the Ideal Gas Law equation, \(PV=nRT\), we'll solve for n: n = \(\frac{PV}{RT}\) Now plug in the values: \(n = \frac{0.901 \mathrm{atm} \times 0.875 \mathrm{L}}{0.0821 \mathrm{L \times atm/mol \times K} \times 308.15 \mathrm{K}}\) After calculating, we get the number of moles: n ≈ 0.0363 mol

Since we have the mass and moles of the gas, we can calculate its molar mass: Molar mass = \(\frac{mass}{moles}\) Molar mass = \(\frac{2.50 \mathrm{g}}{0.0363 \mathrm{mol}}\) After calculating, we get the molar mass: Molar mass ≈ 68.87 g/mol