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

Short answer

(a) The density of NO2 gas at 0.970 atm and 35°C is approximately 1.769 g/L. (b) The molar mass of the unknown gas is approximately 73.962 g/mol.

Step-by-step solution

(a) Calculate the density of NO2 gas using the Ideal Gas Law equation

First, we need to rearrange the Ideal Gas Law equation (PV=nRT) to find the density of the gas. The density of a gas is obtained by dividing its mass(m) by its volume(V). 1. The number of moles (n) can be written as n=m/M, where m is mass and M is the molar mass. 2. After substituting n=m/M in the Ideal Gas Law equation, we have \(P \cdot V = \frac{m}{M} \cdot R \cdot T \). 3. Now, we can get the formula for density, which is given by: Density(\(\rho\)) = \(\frac{m}{V} = \frac{P \cdot M}{R \cdot T}\). Now, we can plug in the given values: Pressure(P) = 0.970 atm Temperature(T) = 35°C = 308.15 K (convert to Kelvin by adding 273.15) Molar mass of NO2 (M) = 46.0055 g/mol (14.0067g/mol for N and 15.999g/mol for O) To carry out the calculations, we will use the value of R corresponding to the units of pressure: R = 0.0821 L atm K⁻¹ mol⁻¹.

Calculate the density of NO2 gas

Now that we have the equation and all the necessary variables, we can calculate the density of NO2 gas. \(\rho = \frac{P \cdot M}{R \cdot T} = \frac{(0.970\,atm) \cdot (46.0055\,g/mol)}{(0.0821\,L\,atm\,K^{-1}\,mol^{-1})(308.15\,K)}\) \(\rho \approx 1.769\,g/L\) The density of NO2 gas at 0.970 atm and 35°C is approximately 1.769 g/L.

(b) Calculate the molar mass of the unknown gas using the Ideal Gas Law equation

To calculate the molar mass of the unknown gas, we will first use the Ideal Gas Law equation (PV=nRT) and rearrange it to find the number of moles (n). Given: Pressure(P) = 685 torr = 0.901 atm (convert to atm by dividing by 760) Temperature(T) = 35°C = 308.15 K (convert to Kelvin by adding 273.15) Volume(V) = 0.875 L Mass(m) = 2.50 g Again, we will use R = 0.0821 L atm K⁻¹ mol⁻¹.

Calculate the number of moles

Using the Ideal Gas Law equation, we can solve for the number of moles (n) of the gas: \(PV=nRT \Rightarrow n = \frac{PV}{RT} = \frac{(0.901\,atm) \cdot (0.875\,L)}{(0.0821\,L\,atm\,K^{-1}\,mol^{-1})(308.15\,K)}\) n ≈ 0.0338 mol

Calculate the molar mass of the unknown gas

Now that we have the number of moles, we can find the molar mass (M) of the gas by using the formula M = m/n: M = (2.50 g)/(0.0338 mol) ≈ 73.962 g/mol The molar mass of the unknown gas is approximately 73.962 g/mol.

Key concepts

Gas Density Calculation

When you're figuring out the density of a gas, the Ideal Gas Law can be your best friend. This law describes how a gas behaves under various physical conditions. For density, you'll use the rearranged equation: \( \rho = \frac{P \cdot M}{R \cdot T} \). Here, \( \rho \) stands for density, \( P \) is pressure, \( M \) is molar mass, \( R \) is the ideal gas constant, and \( T \) is temperature in Kelvin.

To tackle a problem like calculating the density of \( \text{NO}_2 \) gas, you plug the known values into the equation. For example, assume pressure \( (P) \) is 0.970 atm, the molar mass \( (M) \) of \( \text{NO}_2 \) is 46.0055 g/mol, the gas constant \( (R) \) is 0.0821 L atm K⁻¹ mol⁻¹, and temperature \( (T) \) is 308.15 K. After inserting these values, you can find the density to be approximately 1.769 g/L.

Remember: converting temperature to Kelvin is crucial every time you use gas laws.

Molar Mass Determination

The molar mass of a gaseous substance can be determined using the Ideal Gas Law. It links different properties of the gas, including pressure, volume, temperature, and the number of moles. Once you have the number of moles, you can easily calculate the molar mass using the formula: \( M = \frac{m}{n} \). Here \( M \) is the molar mass, \( m \) is mass, and \( n \) is the number of moles.

To find the molar mass of an unknown gas occupying 0.875 L at 0.901 atm and 308.15 K, you'd first determine the moles using the rearranged Ideal Gas Law: \( n = \frac{PV}{RT} \). Substituting in the values, you calculate \( n \approx 0.0338 \) moles. Now, use the mass 2.50 g and divide it by the number of moles: \( M = \frac{2.50}{0.0338} \approx 73.962 \text{ g/mol} \).

This gives you the molar mass of the unknown gas, illustrating the key role the Ideal Gas Law plays in such calculations.

Temperature Conversion

Converting temperature to Kelvin is a crucial step when working with gas laws. The Kelvin scale is essential because it starts at absolute zero, the point at which all molecular motion stops. The conversion is simple: add 273.15 to the temperature in degrees Celsius.

In gas law problems, you always need to use Kelvin. For example, converting 35°C to Kelvin, you get \( 35 + 273.15 = 308.15 \text{ K} \). Without this conversion, calculations like those using the Ideal Gas Law won't be accurate, as the Kelvin scale ensures direct proportionality in equations.

So, always double-check your temperature units are in Kelvin when solving gas-related problems.