Problem 49 - Chemistry Central Science

Question

(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 \(35^{\circ} \mathrm{C}\).

Step-by-Step Solution

Answer

\( a) \) The density of NO₂ gas at 0.970 atm and 35°C is 1.77 g/L. \( b) \) The molar mass of the unknown gas is 73.1 g/mol.

Step 1: Part A: Calculate the moles of NO₂

\( P = 0.970 \ atm \\ T = 35 + 273.15 = 308.15 \ K \\ R = 0.0821 \frac{L \cdot atm}{mol \cdot K} \) Now we can use the molar mass of NO2: M(NO2) = 14.01 (for N) + 2 * 16.00 (for two O) = 46.01 g/mol. We can find the density using the formula: \[ d = \frac{MP}{RT} \] Where d is the density, M is the molar mass, P is the pressure, R is the ideal gas constant, and T is the temperature in Kelvin.

Step 2: Calculate the density

Substitute the known values into the formula: \[ d = \frac{46.01 \frac{g}{mol} \cdot 0.970 \ atm}{0.0821 \frac{L \cdot atm}{mol \cdot K} \cdot 308.15 \ K} \] Now, calculate the density: \[ d = 1.77 \frac{g}{L} \] Thus, the density of NO₂ gas at 0.970 atm and 35°C is 1.77 g/L.

Step 3: Part B: Calculate the moles of the unknown gas

For part (b), we are given the mass (m), volume (V), pressure (P), and temperature (T) of the unknown gas: \( m = 2.50 \ g \\ V = 0.875 \ L \\ P = 685 \ torr = 685 \frac{760 \ atm}{1 \ torr} = 0.901 \ atm \\ T = 35 + 273.15 = 308.15 \ K \\ R = 0.0821 \frac{L \cdot atm}{mol \cdot K} \) Use the Ideal Gas Law equation: \[ PV = nRT \] Solve for n: \[ n = \frac{PV}{RT} \]

Step 4: Substitute the values

Substitute the given values into the formula: \[ n = \frac{(0.901 \ atm)(0.875 \ L)}{(0.0821 \frac{L \cdot atm}{mol \cdot K})(308.15 \ K)} \] Calculate the number of moles: \[ n = 0.0342 \ mol \]

Step 5: Calculate the molar mass

Now, we can use the mass and number of moles to find the molar mass (M) of the gas: \[ M = \frac{m}{n} \] Substitute the known values: \[ M = \frac{2.50 \ g}{0.0342 \ mol} \] Determine the molar mass: \[ M = 73.1 \frac{g}{mol} \] Hence, the molar mass of the unknown gas is 73.1 g/mol.