Equal volumes of monoatomic and diatomic gases of same initial temperature and pressure are mixed.What will be the ratio of sprcific heats of the mixture(Cp/Cv)?
Internal energy may be expressed as
U = ν•R•T/(γ-1), and
U = ν• (i/2) •R•T,
where ν is the amount of substance, i is the degrees of freedom (i1 =3, i2 =5),
γ =Cp/Cv -adiabatic index.
From
p• V = ν•R•T, if V1=V2, then ν1= ν2 = ν.
U= U1+U2,
U = ν•R•T/(γ-1) = (ν1+ ν2) •R•T/(γ-1) =2 ν•R•T/(γ-1),
U1+U2 = ν1• (i1/2) •R•T + ν2• (i2/2) •R•T= (i1+i2) •v•R•T/2.
2 ν•R•T/(γ-1) = (i1+i2) •v•R•T/2.
2/(γ-1) =( i1+i2)/2=(3+5)/2 =4
2γ = 3,
γ =1.5
To determine the ratio of specific heats (Cp/Cv) for a mixture of monoatomic and diatomic gases with equal volumes, we need to consider their molecular properties.
1. Monoatomic gas: A monoatomic gas consists of individual atoms that are not bonded together. Helium (He) and argon (Ar) are examples of monoatomic gases. The specific heat at constant volume (Cv) for a monoatomic gas is 3R/2, where R is the ideal gas constant (8.314 J/(mol·K)).
2. Diatomic gas: A diatomic gas consists of molecules that contain two atoms bonded together. Examples of diatomic gases are nitrogen (N2), oxygen (O2), and hydrogen (H2). The specific heat at constant volume (Cv) for a diatomic gas is 5R/2.
Now, let's find the ratio of specific heats for the mixture:
1. Since the volumes of monoatomic and diatomic gases are equal, the total volume of the mixture will also be twice the volume of one gas.
2. Since the initial temperature and pressure are the same for both gases, we can assume the gases are at thermal equilibrium.
3. To find the ratio of specific heats for the mixture, we can use the principle of equipartition of energy. According to this principle, the total internal energy of the mixture can be expressed as:
Total internal energy = (N₁Cv₁ + N₂Cv₂) , where N₁ and N₂ are the number of moles of each gas, and Cv₁ and Cv₂ are the specific heats at constant volume for the monoatomic and diatomic gases, respectively.
4. Since the volume is the same and the gases are at the same initial temperature and pressure, the number of moles for both gases will also be the same.
5. Therefore, we can simplify the equation as:
Total internal energy = 2N(Cv₁ + Cv₂)
6. The specific heat at constant pressure (Cp) for any ideal gas can be related to its specific heat at constant volume (Cv) by the equation:
Cp = Cv + R
where R is the ideal gas constant.
7. Substituting this relation into the equation for total internal energy, we have:
Total internal energy = 2N[Cv₁ + (Cv₂ + R)]
8. The ratio of specific heats for the mixture (Cp/Cv) can be obtained by using the relation:
Cp/Cv = (Total internal energy)/(R*N)
9. Substituting the values into the equation for total internal energy, we get:
Cp/Cv = [2N(Cv₁ + Cv₂)]/(R*N)
10. Simplifying further, we find:
Cp/Cv = [2(Cv₁ + Cv₂)]/R
11. Substitute the values of Cv₁ = 3R/2 and Cv₂ = 5R/2:
Cp/Cv = [2(3R/2 + 5R/2)]/R
12. Simplifying the expression:
Cp/Cv = (3R + 5R)/R
Cp/Cv = 8
Therefore, the ratio of specific heats (Cp/Cv) for the mixture of equal volumes of monoatomic and diatomic gases is 8.