States of Matter: Question 8

Worked solution

Part (a): Why the two flasks contain equal amounts of gas

Rearranging the ideal gas equation for the amount of substance: $n = \frac{pV}{RT}$

This shows that $n$ depends only on the pressure $p$, the volume $V$, the gas constant $R$ and the temperature $T$. It does not depend on which gas is present. Since flasks 1 and 2 have the same volume ($500\ \text{cm}^3$), the same pressure ($1.00\times10^{5}\ \text{Pa}$) and the same temperature ($298\ \text{K}$), and $R$ is a universal constant, the value of $n = pV/(RT)$ must be identical for both flasks, even though they contain chemically different gases.

Part (b): Calculating the amount of gas in each flask

Converting the volume into SI units (since $1\ \text{m}^3 = 10^6\ \text{cm}^3$): $V = 500\ \text{cm}^3 = \frac{500}{10^6}\ \text{m}^3 = 5.00\times10^{-4}\ \text{m}^3$

Substituting into $n = \dfrac{pV}{RT}$ with $p = 1.00\times10^{5}\ \text{Pa}$ and $T = 298\ \text{K}$: $n = \frac{(1.00\times10^{5})\times(5.00\times10^{-4})}{8.31\times298}$

Working out the numerator and denominator separately: $pV = 1.00\times10^{5}\times5.00\times10^{-4} = 50.0$ $RT = 8.31\times298 = 2476.38$

So: $n = \frac{50.0}{2476.38} = 0.020191\ \text{mol}$

(Check: $2476.38\times0.0202 \approx 50.03$, confirming $n \approx 0.0202\ \text{mol}$.)

To 3 significant figures, $n = 0.0202\ \text{mol}$ in each flask, as explained in part (a).

Part (c): Calculating the masses and explaining the difference

Using $m = n\times M_r$ for each gas, with $n = 0.020191\ \text{mol}$ (unrounded value used to avoid early rounding error):

Oxygen, O₂ ($M_r = 32.0$): $m(\text{O}_2) = 0.020191\times32.0 = 0.6461\ \text{g} \approx 0.646\ \text{g}$

Carbon dioxide, CO₂ ($M_r = 44.0$): $m(\text{CO}_2) = 0.020191\times44.0 = 0.8884\ \text{g} \approx 0.888\ \text{g}$

Explanation: from part (a), both flasks contain exactly the same amount (in mol) of gas, and therefore the same number of molecules. However, each $\text{CO}_2$ molecule has a greater mass than each $\text{O}_2$ molecule, because $M_r(\text{CO}_2) = 44.0$ is greater than $M_r(\text{O}_2) = 32.0$. Since mass $=$ amount $\times$ molar mass, the same number of moles (and molecules) of a heavier substance has a greater total mass, so flask 2 contains more mass of gas than flask 1, even though the two flasks contain equal numbers of gas particles.

Final answers

• (a) $n = pV/(RT)$ depends only on $p$, $V$, $R$ and $T$, which are identical for both flasks, so $n$ must be equal for both, regardless of the gas
• (b) $n = 0.0202\ \text{mol}$ in each flask (3 s.f.)
• (c) mass of O₂ $= 0.646\ \text{g}$; mass of CO₂ $= 0.888\ \text{g}$; equal moles of a heavier gas ($\text{CO}_2$) give a greater total mass than the same moles of a lighter gas ($\text{O}_2$)