Chapter 1, Problem 1.21E

Pressures of gases in mixtures are referred to as partial pressures and are additive. 1.00 L of He gas at 0.75 atm is mixed with 2.00 L of Ne gas at 1.5 atm at a temperature of 25.0 °C to make a total volume of 3.00 L of a mixture. Assuming no temperature change and that He and Ne can be approximated as ideal gases, what are (a) the total resulting pressure, (b) the partial pressures of each component, and (c) the mole fractions of each gas in the mix?

Interpretation Introduction

(a)

Interpretation:

Pressure of gases in mixtures are referred to as partial pressures are additive. 1.00 L of He gas at 0.75 atm is mixed with 200 L of Ne gas at 1.5 atm at a temperature of 25.0 °C to make a total volume of 3.00 L of a mixture. Assuming no temperature change and that He and Ne can be approximated as ideal gases, (a) the total resulting pressure is to be calculated.

Concept introduction:

In physical sciences, the pressure is defined as the perpendicular force acting per unit area (or) the stress at a point within a confined fluid. The symbol for it is p or P. Dalton's law of partial pressure states the fact that the total pressure of a mixture of gases is equal to the sum of the partial pressures of the individual gases in the mixture. The mole fraction ( X ) of any component of a mixture is the ratio of the number of moles of that component to the total number of moles of all the species present in the mixture. Thus, the composition of a gas mixture can be defined by the mole fractions of the gases present.

Answer to Problem 1.21E

The total resulting pressure of ideals gases is ‘1.25 atm’.

Explanation of Solution

Pressure can be defined as a force acting on per unit area i.e.,

P = F/A,

Where, F = force,

A = area on which the force is acted.

Pressure can be conveniently measured for gases and fluids. The pressure of a liquid or a gas is equal to the density of that liquid multiplied by the acceleration due to gravity and the height of the liquid above the certain point. The standard unit for pressure is Pascal (Pa). Other units of pressure, such as pounds per square inch and bar, are also commonly used. The pressure units can also be expressed in grams-force or kilograms-force per square centimeter, without properly identifying the force units. The standard atmosphere (atm) is approximately equal to typical air pressure at earth mean sea level and is having the value of 101325 Pa. Since, Pressure is most prevalently measured by its tendency to displace a column of liquid in a manometer, it is also expressed as depth of a fluid. For example: Centimeters of water (or) millimeters of mercury.

1 Pa = 1N/m² = 1 Kg/ms²

The pressure can be calculated as

Pressure = density of a liquid x acceleration due to gravity x height of liquid in the column

P = ρgh……………………………………………………………………(1)

P = Pressure in Pa

ρ = Density of the liquid in Kg/m³

g = acceleration due to gravity which is given by 9.80 m/s²

h = height of a column in m

Total pressure of a ideal gas mixture is given by individual gas partial pressures. i.e., The total resulting pressure of ideals gases in the above gas mixtures is the summation of Partial pressure of Ne and Partial pressure of He.

Total Pressure of ideals gases = Partial pressure of Ne + Partial pressure of He…………..(2)

Given,

Volume of He gas = 1.00 L

Pressure of He gas = 0.75 atm

Volume of Ne gas = 2.00 L

Pressure of Ne gas = 1.5 atm

Temperature of system = 25 °C = 298 K

Total volume of gas mixture = 3.00 L

From the partial pressure calculations of He and Ne we know that partial pressure of He is 0.25 atm and partial pressure of Ne is 1 atm. Substituting the partial pressure values in equation (2), we get the total pressure as,

Total Pressure of ideals gases = Partial pressure of Ne + Partial pressure of He

= 1 atm + 0.25 atm

= 1.25 atm

Conclusion

Thus, the total resulting pressure of ideal gas mixture is 1.25 atm.

Interpretation Introduction

(b)

Interpretation:

Pressure of gases in mixtures are referred to as partial pressures are additive. 1.00 L of He gas at 0.75 atm is mixed with 200 L of Ne gas at 1.5 atm at a temperature of 25.0 °C to make a total volume of 3.00 L of a mixture. Assuming no temperature change and that He and Ne can be approximated as ideal gases, (a) the total resulting pressure is to be calculated.

Concept introduction:

In physical sciences, the pressure is defined as the perpendicular force acting per unit area (or) the stress at a point within a confined fluid. The symbol for it is p or P. Dalton's law of partial pressure states the fact that the total pressure of a mixture of gases is equal to the sum of the partial pressures of the individual gases in the mixture. The mole fraction ( X ) of any component of a mixture is the ratio of the number of moles of that component to the total number of moles of all the species present in the mixture. Thus, the composition of a gas mixture can be defined by the mole fractions of the gases present.

Answer to Problem 1.21E

The partial pressure of each component in the ideal gas mixtures is calculated as 1) partial pressure of He is 0.25 atm; 2) partial pressure of Ne is 1 atm.

Explanation of Solution

Pressure can be defined as a force acting on per unit area i.e.,

P = F/A,

Where, F = force,

A = area on which the force is acted.

Dalton's law of partial pressure states the fact that the total pressure of a mixture of gases is equal to the sum of the partial pressures of the individual gases in the mixture. In a mixture of gases, each gas contributes to the total pressure of the mixture and this kind of contribution is denoted as partial pressure. In other words, the partial pressure is the pressure the gas if the gas were in the same volume and temperature by itself. Boyle’s law is described as ‘In a closed system, the absolute pressure exerted by a given mass of an ideal gas is inversely proportional to the volume it occupies if the temperature and amount of gas remain unchanged’.

P Α 1/V…………………………………………………………………..(1)

In other words, at fixed value of number of moles of gas (n)

P. V = F(T)………………………………………………………………(2)

Where,

P = Pressure of the gas

V= Volume of the gas

This equation can also be represented as the product of pressure (P) and volume (V) is a constant for a given mass of gas and this holds if the temperature is constant. On comparing the same substance under two different sets of conditions, the gas law can be modified as,

P₁V₁ = P₂V_{2………………………………………………………………………………………….}(3)

* partial pressure of He is calculated as,

Given,

Volume of He gas = 1.00 L = V₁

Pressure of He gas = 0.75 atm = P₁

Temperature of system = 25 °C = 298 K

Total volume of gas mixture = 3.00 L = V₂

Partial Pressure of He = ? = P₂

Substituting the values in equation (3), we get the partial pressure of He as,

P₂ = P₁V₁/V₂

= 0.75 atm x 1.00 L / 3.00 L

∴P₂ = 0.25 atm

* partial pressure of Ne is calculated as,

Volume of Ne gas = 2.00 L = V₁

Pressure of Ne gas = 1.5 atm = P₁

Total volume of gas mixture = 3.00 L = V₂

Partial Pressure of Ne = ? = P₂

Substituting the values in equation (3), we get the partial pressure of Ne as,

P₂ = P₁V₁/V₂

= 2 L x 1.5 atm / 3.00 L

∴P₂ = 1 atm

Conclusion

The partial pressure of He is 0.25 atm and partial pressure of Ne is 1 atm.

Interpretation Introduction

(c)

Interpretation:

Pressure of gases in mixtures are referred to as partial pressures are additive. 1.00 L of He gas at 0.75 atm is mixed with 200 L of Ne gas at 1.5 atm at a temperature of 25.0 °C to make a total volume of 3.00 L of a mixture. Assuming no temperature change and that He and Ne can be approximated as ideal gases, (a) the total resulting pressure is to be calculated.

Concept introduction:

In physical sciences, the pressure is defined as the perpendicular force acting per unit area (or) the stress at a point within a confined fluid. The symbol for it is p or P. Dalton's law of partial pressure states the fact that the total pressure of a mixture of gases is equal to the sum of the partial pressures of the individual gases in the mixture. The mole fraction ( X ) of any component of a mixture is the ratio of the number of moles of that component to the total number of moles of all the species present in the mixture. Thus, the composition of a gas mixture can be defined by the mole fractions of the gases present.

Answer to Problem 1.21E

The mole fraction of He is 0.2 and the mole fraction of Ne is 0.8.

Explanation of Solution

The mole fraction ( X ) of any component of a mixture is the ratio of the number of moles of that component to the total number of moles of all the species present in the mixture.

$\begin{array}{l}{\text{X}}_{\text{A}}=\frac{\text{moles of A}}{\text{total}\text{moles}}=\frac{{\text{n}}_{\text{A}}}{{\text{n}}_{\text{total}}}=\frac{{\text{n}}_{\text{A}}}{{\text{n}}_{\text{A}}+{\text{n}}_{\text{B}}}\\ {\text{X}}_{\text{B}}=\frac{\text{moles of B}}{\text{total}\text{moles}}=\frac{{\text{n}}_{\text{B}}}{{\text{n}}_{\text{total}}}=\frac{{\text{n}}_B}{{\text{n}}_{\text{A}}+{\text{n}}_{\text{B}}}\end{array}$

The mole fraction is a dimensionless quantity, which is having values between 0 and 1. For example, if X_A = 0, then there is no A present I the mixture and if X_A = 1, then there is only one component i.e., A only. This trend is also applicable to B in the mixture.

the ideal gas equation P₁V₁ = n₁RT

and n₁ = P₁V₁/RT ………………………………….(4)

* the number of moles and mole fraction of He:

Volume of He gas = 1.00 L = V₁

Pressure of He gas = 0.75 atm = P₁

Temperature of system = 25 °C = 298 K

The gas constant R = 0082 L atm / K. mol

The number of mole of He = n₁ = ?

Substituting the values in equation (4), we get

$\begin{array}{l}{\text{n}}_1=\frac{0.75\text{atm}\times \text{1}\text{.00}\text{L}}{0.0823\text{L atm/}\text{K}\text{mol}\times \text{298}\text{K}}\\ =\frac{0.75}{24.53}\\ {\text{n}}_1=0.03\end{array}$

The number of moles of He is 0.03 and the mole fraction of He (X_He) ;

$\begin{array}{l}{\text{X}}_{\text{He}}=\frac{0.03}{0.03+0.12}\\ {\text{X}}_{\text{He}}=0.2\end{array}$

Thus, the mole fraction of He is 0.2.

* the number of moles and mole fraction of Ne:

Volume of Ne gas = 2.00 L = V₁

Pressure of Ne gas = 1.5 atm = P₁

Temperature of system = 25 °C = 298 K

The gas constant R = 0082 L atm / K. mol

The number of mole of Ne = n₁ = ?

Substituting the values in equation (4), we get

$\begin{array}{l}{\text{n}}_1=\frac{1.5\text{atm}\times \text{2}\text{.00}\text{L}}{0.0823\text{L}\text{.atm/K}\text{.mol}\times \text{298}\text{K}}\\ =\frac{3.0}{24.53}\\ {\text{n}}_1=0.12\end{array}$

The number of moles of Ne is 0.12 and the mole fraction of Ne (X_Ne);

$\begin{array}{l}{\text{X}}_{\text{Ne}}=\frac{0.12}{0.03+0.12}\\ {\text{X}}_{\text{Ne}}=0.8\end{array}$

Thus, the mole fraction of Ne is 0.8.

Conclusion

Thus, the mole fraction of He is 0.2 and the mole fraction of Ne is 0.8.