Sure, let's solve this problem step by step.
First, we need to determine the number of moles of each gas in the mixture.
### Step 1: Calculate the number of moles of each gas
Methane (CH₄):
1. Given mass of CH₄: 4.0 grams
2. Molar mass of CH₄: 16.04 g/mol
To find the number of moles of methane, we use the formula:
[tex]\[
\text{moles of CH₄} = \frac{\text{mass of CH₄}}{\text{molar mass of CH₄}}
\][/tex]
[tex]\[
= \frac{4.0 \text{ grams}}{16.04 \text{ g/mol}} \approx 0.249 \text{ moles of CH₄}
\][/tex]
Helium (He):
1. Given mass of He: 2.0 grams
2. Molar mass of He: 4.00 g/mol
To find the number of moles of helium, we use the formula:
[tex]\[
\text{moles of He} = \frac{\text{mass of He}}{\text{molar mass of He}}
\][/tex]
[tex]\[
= \frac{2.0 \text{ grams}}{4.00 \text{ g/mol}} = 0.5 \text{ moles of He}
\][/tex]
### Step 2: Calculate the total number of moles of the gas mixture
[tex]\[
\text{total moles} = \text{moles of CH₄} + \text{moles of He}
\][/tex]
[tex]\[
= 0.249 \text{ moles of CH₄} + 0.5 \text{ moles of He} \approx 0.749 \text{ moles}
\][/tex]
### Step 3: Determine the total volume of the gas mixture at STP
Given that at STP (273 K and 1.0 atm), one mole of any ideal gas occupies 22.4 liters, we can calculate the volume of the gas mixture using the formula:
[tex]\[
\text{volume at STP} = \text{total moles} \times \text{molar volume at STP}
\][/tex]
[tex]\[
= 0.749 \text{ moles} \times 22.4 \text{ L/mol} \approx 16.786 \text{ L}
\][/tex]
So, the volume of the gas mixture at STP is approximately 16.786 liters.
Therefore, the closest answer to the volume of the gas mixture at STP is 17 L.
