Determine the ratio of the adiabatic index of a gas mixture

To solve the problem, it is necessary to determine the adiabatic index of a mixture of gases obtained by mixing 5 g of helium and 2 g of hydrogen, and compare it with the adiabatic index of pure components.

Let's move on to solving the problem. The adiabatic index is determined by the formula:

γ = Cp / Cv,

where Cp and Cv are, respectively, the heat capacities at constant pressure and constant volume. For pure gases, adiabatic indices can be determined from tables or using the following formulas:

γ(He) = 1.67, γ(H2) = 1.41.

For a mixture of gases, the adiabatic index can be determined by the formula:

γ = (Cp1 + Cp2) / (Cv1 + Cv2),

where Cp1 and Cv1 are the heat capacities at constant pressure and constant volume, respectively, for the first component, and Cp2 and Cv2 for the second component.

For helium and hydrogen, the heat capacities at constant pressure and constant volume can be found in tables or using the following values:

Cp(He) = 20.78 J/(molK), Cv(He) = 12.47 J/(molK), Cp(H2) = 28.83 J/(molK), Cv(H2) = 20.43 J/(molTO).

To find the heat capacity, you can use the following formula:

C = q / (n * ΔT),

where q is the amount of heat transferred to the system, n is the amount of substance, ΔT is the temperature change.

For our mixture of gases, the amount of substance can be found using the formula:

n = m / M,

where m is the mass of the gas mixture, M is the molar mass.

For helium and hydrogen, molar masses can be found in tables or use the following values:

M(He) = 4 g/mol, M(H2) = 2 g/mol.

Now we can calculate the heat capacities for each component:

Cp(He) = q(He) / (n(He) * ΔT), Cv(He) = Cp(He) - R, Cp(H2) = q(H2) / (n(H2) * ΔT), Cv(H2) = Cp(H2) - R,

where R is the universal gas constant. For ease of calculation, you can use the following values:

R = 8.31 J/(molK), R = 0.0821 latm/(mol*K).

Substituting the found values, we get:

Cp(He) = 20.78 J/(molK), Cv(He) = 8.31 J/(molK), Cp(H2) = 28.83 J/(molK), Cv(H2) = 8.4 J/(molTO).

Now we can find the adiabatic exponent for a mixture of gases:

γ = (Cp1 + Cp2) / (Cv1 + Cv2) = (20.78 + 28.83) / (8.31 + 8.4) ≈ 1.66.

The obtained value of the adiabatic index for a mixture of gases is close to the adiabatic index of helium and less than the adiabatic index of hydrogen.

Thus, the ratio of the adiabatic index of a mixture of gases obtained by mixing 5 g of helium and 2 g of hydrogen to the adiabatic index of pure components is approximately 1.66 for the mixture, 1.67 for helium and 1.41 for hydrogen. This suggests that the adiabatic index of a gas mixture is close to the adiabatic index of helium and less than the adiabatic index of hydrogen.

Product Description: Determination of the ratio of the adiabatic index of a gas mixture

This digital product is a solution to the problem of determining the ratio of the adiabatic index of a gas mixture. The solution contains a detailed record of the problem conditions, formulas and laws used in the solution, the derivation of the calculation formula and the answer.

The solution is presented in a convenient and beautifully designed HTML format, which allows you to quickly and easily familiarize yourself with the material and visually evaluate its quality.

This product may be useful to students and teachers studying thermodynamics and gas dynamics, as well as anyone interested in this field of science.

This digital product is a detailed solution to the problem of determining the ratio of the adiabatic index of a gas mixture obtained by mixing 5 g of helium and 2 g of hydrogen to the adiabatic index of pure components. The solution contains a brief record of the conditions of the problem, formulas and laws used in the solution, the derivation of the calculation formula and the answer.

To solve the problem, it is necessary to determine the adiabatic index of a gas mixture using the formula γ = (Cp1 + Cp2) / (Cv1 + Cv2), where Cp1 and Cv1 are the heat capacities at constant pressure and constant volume, respectively, for the first component (helium), and Cp2 and Cv2 - for the second component (hydrogen).

For pure gases, adiabatic indices can be determined from tables or using the following formulas: γ(He) = 1.67, γ(H2) = 1.41. For helium and hydrogen, the heat capacity at constant pressure and constant volume can be found in the tables or use the following values: Cp(He) = 20.78 J/(molK), Cv(He) = 12.47 J/(molK), Cp(H2) = 28.83 J/(molK), Cv(H2) = 20.43 J/(molK).

To find the heat capacity, you can use the formula C = q / (n * ΔT), where q is the amount of heat transferred to the system, n is the amount of substance, ΔT is the change in temperature. For our mixture of gases, the amount of substance can be found using the formula n = m / M, where m is the mass of the mixture of gases, M is the molar mass.

After finding all the necessary values, you can substitute them into the formula γ = (Cp1 + Cp2) / (Cv1 + Cv2) and get the answer. In this case, the adiabatic index for a mixture of gases will be approximately 1.66, which is close to the adiabatic index of helium and less than the adiabatic index of hydrogen.

This product may be useful to students and teachers studying thermodynamics and gas dynamics, as well as anyone interested in this field of science. If you have questions about solving a problem, you can contact the author of the solution for help.

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To determine the ratio of the adiabatic index of a gas mixture obtained by mixing 5 g of helium and 2 g of hydrogen to the adiabatic index of pure components, it is necessary to use the formula for calculating the adiabatic index of a gas:

γ = Cp/Cv,

where γ is the adiabatic exponent, Cp is the heat capacity at constant pressure, and Cv is the heat capacity at constant volume.

To calculate the adiabatic index of a gas mixture, it is necessary to know the adiabatic index of each of the components and their volume fractions in the mixture. Since the masses of the components are indicated in the problem, it is necessary to first determine their molar masses.

The molar mass of helium is 4 g/mol, and the molar mass of hydrogen is 2 g/mol. Therefore, the number of moles of helium is 5 g / 4 g/mol = 1.25 mol, and the number of moles of hydrogen is 2 g / 2 g/mol = 1 mol. The total number of moles in the mixture is 1.25 mol + 1 mol = 2.25 mol.

The volume fraction of helium in the mixture is (number of moles of helium * molar volume of helium) / (total number of moles * molar volume of the mixture) = (1.25 mol * 24.79 l/mol) / (2.25 mol * 24.45 l/mol) ≈ 0.570. The volume fraction of hydrogen in the mixture is 1 - 0.570 = 0.430.

The adiabatic index of helium at constant volume is 1.67, and at constant pressure - 1.40. The adiabatic index of hydrogen at constant volume is 1.40, and at constant pressure it is 1.41.

To calculate the adiabatic index of a gas mixture, it is necessary to weightedly average the adiabatic exponents of the components, taking into account their volume fractions in the mixture:

γmixtures = (γhelium * Vhelium + γhydrogen * Vhydrogen) / (Vhelium + Vhydrogen),

where Vhelium and Vhydrogen are the volumes of helium and hydrogen in the mixture, respectively.

The volume of helium is 0.570 * molar volume of the mixture ≈ 13.9 l, and the volume of hydrogen is 0.430 * molar volume of the mixture ≈ 10.3 l.

Now you can substitute the values ​​into the formula and calculate the adiabatic index of a gas mixture:

γsmesi = (1.67 * 13.9 L + 1.40 * 10.3 L) / (13.9 L + 10.3 L) ≈ 1.58.

Answer: the ratio of the adiabatic index of a gas mixture obtained by mixing 5 g of helium and 2 g of hydrogen to the adiabatic index of pure components is 1.58 / 1.67 ≈ 0.946 for helium and 1.58 / 1.41 ≈ 1.12 for hydrogen.

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