States of Matter: Question 4

Worked solution

Step 1: Rearrange the ideal gas equation

The ideal gas equation is: $pV = nRT$

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

Step 2: Convert all quantities into SI units

The ideal gas equation requires pressure in pascals (Pa), volume in cubic metres ($\text{m}^3$), and temperature in kelvin (K).

Pressure: since $1\ \text{kPa} = 10^3\ \text{Pa}$, $p = 250\ \text{kPa} = 2.50\times10^{5}\ \text{Pa}$

Volume: since $1\ \text{m}^3 = 10^3\ \text{dm}^3$, $V = 2.00\ \text{dm}^3 = 2.00\times10^{-3}\ \text{m}^3$

Temperature: the temperature is already given in kelvin, so no conversion is needed: $T = 500\ \text{K}$

Step 3: Substitute and calculate

$n = \frac{pV}{RT} = \frac{(2.50\times10^{5})\times(2.00\times10^{-3})}{8.31\times500}$

Working out the numerator and denominator separately: $pV = 2.50\times10^{5}\times2.00\times10^{-3} = 500$ $RT = 8.31\times500 = 4155$

So: $n = \frac{500}{4155} = 0.1203\ \text{mol}$

To 3 significant figures, $n = 0.120\ \text{mol}$.

Why the other options are wrong

• B ($120\ \text{mol}$): this results from forgetting to convert the volume from dm³ to m³ (using $V = 2.00$ directly), which makes $n$ exactly 1000 times too large.
• C ($1.20\times10^{-4}\ \text{mol}$): this results from forgetting to convert the pressure from kPa to Pa (using $p = 250$ directly), which makes $n$ exactly 1000 times too small.
• D ($8.31\ \text{mol}$): this comes from inverting the rearrangement of the ideal gas equation and calculating $\frac{RT}{pV} = \frac{4155}{500} = 8.31$ instead of $\frac{pV}{RT}$.

Final answer

• The amount of gas present is $\boxed{0.120}$ mol, option A.