Find the specific heat capacities cp and cv for gas molecules, e

Finding the specific heat capacities cp and cv for gas molecules
Given: The probable speed of movement of gas molecules under normal conditions is 484.5 m/s The speed of sound is 388 m/s
Find: specific heat capacities cp and cv for gas molecules

Answer:

To solve the problem we use the following formulas:

Speed of sound in gas: c = sqrt(γ * R * T / M), where γ is the adiabatic exponent, R is the universal gas constant, T is the absolute temperature, M is the molar mass of the gas
The relationship between specific heat capacities: сp - сv = R/M

From the relationship between specific heat capacities we obtain:

сp = сv + R/M

To find the specific heat capacities cv and cp, it is necessary to find the adiabatic index γ. To do this, we use the formula for the speed of sound in gas:

c = sqrt(γ * R * T / M)

As M we take the molecular weight of air M = 29 g/mol, because it is close to the molecular weight of most gases. Then:

γ = c^2 * M / R / T = (388 m/s)^2 * 29 g/mol / (8.31 J/mol*K * 273 K) ≈ 1.4

Now we can find the specific heat capacities:

сv = R/(γ - 1)/M ≈ 0.718 J/g*K

сp = γ * R /(γ - 1)/M ≈ 1.005 J/g*K

Answer: specific heat capacity at constant volume cv ≈ 0.718 J/g*K, specific heat capacity at constant pressure cp ≈ 1.005 J/g*K

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To find the specific heat capacities cp and cv for gas molecules, it is necessary to use the Mayer equation:

c = сp - сv,

where c is the speed of sound in gas, cp is the specific heat capacity at constant pressure, cv is the specific heat capacity at constant volume.

First, let's find the ratio of the specific heat capacities of the gas:

γ = сп / св.

For a monoatomic gas such as He or Ne, γ = 5/3. For a diatomic gas such as O2 or N2, γ = 7/5.

In our case, we do not know what gas is being considered, so we will use the general formula:

γ = 1 + 2 / f,

where f is the degree of freedom of gas molecules. For a diatomic gas f = 5, for a monatomic gas - f = 3.

Let us determine the degree of freedom of gas molecules, knowing that its probable speed of movement under normal conditions is 484.5 m/s:

v = √(3kT/m), where k is Boltzmann’s constant, T is the gas temperature, m is the mass of the gas molecule.

Let us express the temperature of the gas:

T = m * v^2 / 3k.

It is also known that the speed of sound propagation in gas is 388 m/s:

с = √(γ * p / ρ),

where p is the gas pressure, ρ is the gas density.

Let us express the gas pressure:

p = ρ * с^2 / c.

Since we do not know the density of the gas, we cannot determine the specific heat capacities cp and cv.

To solve the problem, additional information about the gas is required, for example, its molecular weight or density under normal conditions. Without such information, solving the problem is impossible.

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