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Learning Goal:

To use partial pressures in a mixture of gases, the total pressure of the gas mixture is equal to the sum of the partial pressures of the individual gases. For example, if you have a mixture of helium at 2 atm and argon at 4 atm, then the total pressure is 6 atm.

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Question:

A volume of 18.0 L contains a mixture of 0.250 mole [tex]$N_2$[/tex], 0.250 mole [tex]$O_3$[/tex], and an unknown quantity of He. The temperature of the mixture is [tex]$0^{\circ}C$[/tex], and the total pressure is [tex][tex]$1.00 \, atm$[/tex][/tex]. How many grams of helium are present in the gas mixture?

Express your answer to three significant figures and include the appropriate units.

Answer:
mass of [tex]$He =$ \square$[/tex]



Answer :

GinnyAnswer
To solve this problem, we need to determine the mass of helium (He) present in the gas mixture. We will use the Ideal Gas Law and the concept of partial pressures step-by-step. Here is the detailed solution:

1. Given Parameters:
- Volume (V) = 18.0 L
- Moles of N₂ (n_N2) = 0.250 moles
- Moles of O₂ (n_O2) = 0.250 moles
- Total pressure (P_total) = 1.00 atm
- Temperature (T) = 0°C
- Universal gas constant (R) = 0.0821 L·atm·K⁻¹·mol⁻¹

2. Convert Temperature to Kelvin:
- Temperature in Kelvin (T_K) = 0 + 273.15 = 273.15 K

3. Calculate Total Moles of Gases Present (excluding He):
- Total moles of gases (excluding He) = moles of N₂ + moles of O₂
- Total moles = 0.250 + 0.250 = 0.500 moles

4. Using the Ideal Gas Law to calculate the total moles and solve for the moles of He:
- Ideal Gas Law: [tex]\( PV = nRT \)[/tex]
- Rearrange to solve for total moles (including He):
[tex]\[ n = \frac{P \cdot V}{R \cdot T} \][/tex]
- Substituting the values:
[tex]\[ n_{total} = \frac{1.00 \: \text{atm} \cdot 18.0 \: \text{L}}{0.0821 \: \text{L·atm·K⁻¹·mol⁻¹} \cdot 273.15 \: \text{K}} = 0.8026535727113838 \: \text{moles} \][/tex]
- Subtracting the moles of N₂ and O₂ to find the moles of He:
[tex]\[ n_{He} = 0.8026535727113838 \: \text{moles} - 0.500 \: \text{moles} = 0.3026535727113838 \: \text{moles} \][/tex]

5. Calculate the Mass of Helium:
- Molar mass of Helium (He) = 4.00 g/mol
- Mass of Helium:
[tex]\[ \text{mass of He} = \text{moles of He} \times \text{molar mass of He} = 0.3026535727113838 \: \text{moles} \times 4.00 \: \text{g/mol} = 1.2106142908455353 \: \text{g} \][/tex]

- Rounded to three significant figures: mass of He ≈ 1.21 g

Thus, the mass of helium (He) present in the gas mixture is approximately [tex]\( 1.21 \)[/tex] grams.

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