Question

A large flask is evacuated and weighed, filled with argon gas, and then reweighed. When reweighed, the flask is found to have gained \(3.224 \mathrm{~g}\). It is again evacuated and then filled with a gas of unknown molar mass. When reweighed, the flask is found to have gained \(8.102\) g. (a) Based on the molar mass of argon, estimate the molar mass of the unknown gas. (b) What assumptions did you make in arriving at your answer?

Short answer

The estimated molar mass of the unknown gas is 100.4 g/mol, based on the molar mass of argon and the assumption that both argon gas and the unknown gas behave like ideal gases under the same temperature and pressure conditions.

Step-by-step solution

Find moles of argon gas

Given the mass of argon gained by the flask, we can determine the number of moles of argon gas using the molar mass of argon (Ar). The molar mass of argon is approximately 39.95 g/mol. Use the formula: \(n_{Ar} = \frac{m_{Ar}}{M_{Ar}}\) Where: \(n_{Ar}\) = moles of argon gas \(m_{Ar}\) = mass of argon gas (3.224 g) \(M_{Ar}\) = molar mass of argon (39.95 g/mol) So, the moles of argon gas is: \(n_{Ar} = \frac{3.224}{39.95} \approx 0.0807\) moles

Find the volume of the flask

We will assume that argon gas behaves like an ideal gas. So, we can apply the ideal gas law to find the volume of the flask. The ideal gas law is given by: \(PV = nRT\) Where: P = pressure V = volume n = moles of gas R = ideal gas constant (8.314 J/mol·K) T = temperature in Kelvin We are not given temperature and pressure values, but we can assume that the flask is at the same temperature and pressure for both argon and the unknown gas. Since we are not using the absolute values of P and T, we can simply represent the volume of the flask as a ratio of moles of the gas and set it equal to a constant (k): \(V = kn_{Ar}\) Substituting the value of argon moles: \(V = k\cdot 0.0807\)

Find the moles of unknown gas

We are given the mass of the unknown gas gained by the flask (8.102 g). To find the moles of the unknown gas, let's denote its molar mass as \(M_x\): \(n_x = \frac{m_x}{M_x}\) Where: \(n_x\) = moles of the unknown gas \(m_x\) = mass of the unknown gas (8.102 g) \(M_x\) = molar mass of the unknown gas

Apply the ideal gas law to estimate the molar mass of the unknown gas

Now we have the volume of the flask and the number of moles of argon. We can apply the ideal gas law and the relationship between the moles of the unknown gas and the volume of the flask: \(V = kn_x\) We can combine the expressions for the volume of the flask for argon and the unknown gas: \(k\cdot 0.0807 = k\cdot n_x\) Solving for the moles of the unknown gas: \(n_x = 0.0807\) Now we can find the molar mass of the unknown gas: \(M_x = \frac{m_x}{n_x} = \frac{8.102}{0.0807} \approx 100.4 \mathrm{~g/mol}\) So, the estimated molar mass of the unknown gas is 100.4 g/mol.

Discuss the assumptions

(a) In our calculations, we assumed that both argon gas and the unknown gas behave like ideal gases. (b) We also assumed that the flask is at the same temperature and pressure for both argon and the unknown gas. If any of these assumptions are not met, the estimated molar mass of the unknown gas might not be accurate.