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Boyle's Law is a key concept in understanding how gases behave under constant temperature conditions. It states that the pressure of a gas is inversely proportional to its volume, as long as the temperature remains constant. This means if you increase the volume, the pressure decreases, and vice versa.
In the exercise given, when the flask is warmed to 490 K with the stopcock open, the gas expands. According to Boyle's Law, the volume remains constant since it's a flask, but the gas escapes to equalize with atmospheric pressure, which is 1.013 x 10鈦� Pa. Boyle's Law helps us understand this process because the changes in pressure and gas escaping are directly related to the volumes available inside and outside the flask.
Once the stopcock is closed and the flask begins to cool back to its original temperature, Boyle鈥檚 Law is not directly applied anymore since the conditions then involve changing temperatures. However, the initial escape of gas under constant temperature conditions can be best explained with Boyle鈥檚 Law, as it lays the groundwork for understanding how pressure-volume relationships adjust.

Gas pressure calculations are central to analyzing the behavior of gases in enclosed systems. The Ideal Gas Law, expressed as \( PV = nRT \), is essential for calculating the relationships between pressure (P), volume (V), temperature (T), and the amount (n) or moles of gas, where \( R \) is the ideal gas constant.
In the given exercise, we initially use this formula to determine the number of moles of ethane present at the original conditions: \( n = \frac{PV}{RT} \). Using \( P = 1.013 \times 10^5 \) Pa, \( V = 1.5 \times 10^{-3} \) m鲁, \( R = 8.314 \) J/mol K, and \( T = 300 \) K, we find that the moles of ethane initially present equal approximately 0.061 mol.
After the warming and gas escape with the stopcock open, the pressure is re-calculated once the system returns to its original temperature of 300 K but with decreased moles remaining. The same ideal gas law returns a final pressure of about 0.94 x 10鈦� Pa. This shows how gas pressure changes based on moles present and the constant volume of the system.

The molar mass of a substance is the mass of one mole of that substance, typically expressed in grams per mole. For ethane \((\text{C}_2\text{H}_6)\), the given molar mass is 30.1 g/mol, which is useful for converting between the amount of substance in moles and its mass.
To find the mass of ethane remaining after the sequence of heating with the stopcock open, then cooling while closed, we use the remaining moles determined from the gas law calculations. The final step in the exercise involves multiplying the moles by the molar mass: \( \text{mass} = n \times \text{molar mass} = 0.061 \times 30.1 = 1.84 \) grams.
This computation shows how molar mass connects the amount of substance in chemical terms (moles) to a practical measurement (grams) that can be weighed. Understanding molar mass is crucial for translating theoretical chemistry calculations into laboratory and real-world applications.