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Sagot :
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.
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.
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