To determine the mass of helium remaining in the tank, we'll use the Ideal Gas Law:

$PV = nRT$

Where:

• $P$ = pressure (in Pascals, Pa)
• $V$ = volume (in cubic meters, m³)
• $n$ = number of moles of gas
• $R$ = ideal gas constant $R = 8.314 \, \text{J/mol·K}$
• $T$ = temperature (in Kelvin, K)

Step 1: Convert the given values into SI units

Given:

• Volume, $V = 6 \, \text{m}^3$
• Temperature, $T = 400 \, \text{K}$
• Pressure is evacuated to 740 mm Hg vacuum.

A vacuum of 740 mm Hg means the absolute pressure in the tank is reduced by 740 mm Hg from atmospheric pressure. Atmospheric pressure is 760 mm Hg, so:

$P = 760 \, \text{mm Hg} - 740 \, \text{mm Hg} = 20 \, \text{mm Hg}$

Now, convert $P$ from mm Hg to Pascals:

$P = 20 \, \text{mm Hg} \times 133.322 \, \text{Pa/mm Hg} = 2666.44 \, \text{Pa}$

Step 2: Apply the Ideal Gas Law

Now, solve for the number of moles $n$ using the Ideal Gas Law:

$n = \frac{PV}{RT}$

Substitute the known values:

$n = \frac{(2666.44 \, \text{Pa})(6 \, \text{m}^3)}{(8.314 \, \text{J/mol·K})(400 \, \text{K})}$

Let's calculate $n$.The number of moles of helium remaining in the tank is approximately $4.81 \, \text{mol}$.

Step 3: Calculate the mass of helium

To find the mass of helium, we use the molar mass of helium, which is $M_{\text{He}} = 4.0026 \, \text{g/mol}$. The mass $m$ is:

$m = n \times M_{\text{He}}$

Let's calculate the mass in grams and convert it to kilograms.The mass of helium remaining in the tank is approximately $0.0193 \, \text{kg}$ or 19.3 grams.

Would you like further details or have any additional questions?

Here are 5 related questions to expand on the topic:

1. How would the mass of helium change if the temperature were increased?
2. What if the pressure dropped even further, say to 10 mm Hg?
3. How does the molar mass of a gas affect the calculation of its mass?
4. What happens to the mass if a different gas, like nitrogen, was used instead of helium?
5. How would the calculations change if the volume of the tank were halved?

Tip: Always make sure to convert all units to the SI system before performing calculations, especially in gas laws!