A) The Volume of the vessel is; 250 Litres
B) The partial pressure of H₂ in the vessel if X_he = 0.75 is;
Partial pressure of H₂ = 189 torr
A) We are given;
V₁ = 20 L
P₁ = 25 atm
P₂ = 2 atm
We can find the volume V₂ of the vessel using boyles law which is;
P₁ × V₁ = P₂ × V₂
Making V₂ the subject gives;
V₂ = (P₁ × V₁)/P₂
Plugging in the relevant values gives;
V₂ = (25 × 20)/2
V₂ = 250 L
B) We are given;
Total pressure = 756 torr
Mole fraction of Helium; X_he = 0.75
Since there is a mixture of H₂ and He in the vessel, then;
Mole fraction of H₂ is;
X_H₂ = 1 - X_he
X_H₂ = 1 - 0.75
X_H₂ = 0.25
According to Dalton's Law of partial pressure, the formula for partial pressure is;
Partial pressure of gas = mole fraction of gas × total pressure of gas
Thus;
Partial pressure of H₂ = 0.25 × 756
Partial pressure of H₂ = 189 torr
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Taking into account the Boyle's law and Dalton's partial pressure law:
The gas laws are a set of chemical and physical laws that allow determining the behavior of gases in a closed system. The parameters evaluated in these laws are pressure, volume, temperature and moles.
Pressure and volume are related by Boyle's law, one of the gas laws, which says that the volume occupied by a given mass of gas at constant temperature is inversely proportional to pressure.
Boyle's law is expressed mathematically as:
P×V = k
Now it is possible to assume that you have a certain volume of gas V1 that is at a pressure P1 at the beginning of the experiment. If you vary the volume of gas to a new value V2, then the pressure will change to P2, and it will be fulfilled:
P1× V1 = P2× V2
In this case, you know:
Replacing:
20 L× 25 atm= V2× 2 atm
Solving
[tex]V2=\frac{20 Lx 25 atm}{2 atm}[/tex]
V2= 250 L
In summary, the volume of the vessel is 250 L.
The pressure exerted by a particular gas in a mixture is known as its partial pressure. So, Dalton's law states that the total pressure of a gas mixture is equal to the sum of the pressures that each gas would exert if it were alone:
PT = PA + PB
Dalton's partial pressure law can also be expressed in terms of the mole fraction of the gas in the mixture.
So in a mixture of two or more gases, the partial pressure of gas A can be expressed as:
PA = XA× PT
In this case you know that the total pressure is 756 torr and XHe = 0.75.
Since there is a mixture of H₂ and He in the vessel, then mole fraction of H₂ is calculated as:
[tex]x_{He} +x_{H_{2} } =1[/tex]
[tex]0.75 +x_{H_{2} } =1[/tex]
[tex]x_{H_{2} } =1 - 0.75[/tex]
[tex]x_{H_{2} } =0.25[/tex]
So, the partial pressure of H₂ can be calculated as:
[tex]P_{H_{2} }=x_{H_{2} }P_{T}[/tex]
[tex]P_{H_{2} }=[/tex]0.25×756 torr
[tex]P_{H_{2} }=[/tex]189 torr
Then, the partial pressure of H₂ is 189 torr.
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