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प्रश्न
Derive the ratio of two specific heat capacities of monoatomic, diatomic and triatomic molecules.
Deduce an expression for molar specific heat of a monoatomic gas at constant volume.
उत्तर
Application of the law of equipartition energy in the specific heat of gas, Meyer’s relation CP − CV = R connects the two specific heats for one mole of an ideal gas.
The equipartition law of energy is used to calculate the value of CP – CV and the ratio between them γ = `"C"_"P"/"C"_"V"`. Here γ is called the adiabatic exponent.
(i) Monatomic molecule: Average kinetic energy of a molecule = `[3/2"kT"]`
Total energy of a mole of gas = `3/2"kT" xx "N"_"A" = 3/2"RT"`
For one mole, the molar specific heat at constant volume
CV = `"dU"/"dT" = "d"/"dT"[3/2"RT"]`
CV = `[3/2"R"]`
`"C"_"P" = "C"_"V" + "R" = 3/2"R" + "R" = 5/2"R"`
The ratio of specific heats, γ = `"C"_"P"/"C"_"V" = (5/2"R")/(3/2"R") = 5/3` = 1.67
(ii) Diatomic molecule: Average kinetic energy of a diatomic molecule at low temperature = `5/2"kT"`
Total energy of one mole of gas = `5/2"kT" xx "N"_"A" = 5/2"RT"`
(Here, the total energy is purely kinetic)
For one mole Specific heat at constant volume
CV = `"dU"/"dT" = [5/2"RT"] = 5/2"R"`
But, `"C"_"P" = "C"_"V" + "R" = 5/2"R" + "R" = 7/2"R"`
∴ γ = `"C"_"P"/"C"_"V" = (7/2"R")/(5/2"R") = 7/5` = 1.40
Energy of a diatomic molecule at high temperature is equal to `7/2"RT"`
CV = `"dU"/"dT" = "d"/"dT" [7/2"RT"] = 7/2"R"`
∴ CP = `"C"_"V" + "R" = 7/2"R" + "R"`
CP = `9/2"R"`
Note that the CV and CP are higher for diatomic molecules than the mono atomic molecules. It implies that increasing the temperature of diatomic gas molecules by 1°C requires more heat energy than monoatomic molecules.
∴ γ = `"C"_"P"/"C"_"V" = (9/2"R")/(7/2"R") = 9/7` = 1.28
(iii) Triatomic molecule:
(a) Linear molecule:
Energy of one mole = `7/2"kT" xx "N"_"A" = 7/2"RT"`
CV = `"dU"/"dT" = "d"/"dT"[7/2"RT"]`
CV = `7/2"R"`
CP = `"C"_"P"/"C"_"V" + "R" = 7/2"R" + "R" = 9/2"R"`
∴ γ = `"C"_"P"/"C"_"V" = (9/2"R")/(7/2"R") = 9/7` = 1.28
(b) Non-linear molecule:
Energy of a mole = `6/2"kT" xx "N"_"A" = 6/2"RT"` = 3RT
CV = `"dU"/"dT"` = 3R
CP = `"C"_"V" + "R"` = 3R + R = 4R
∴ γ = `"C"_"P"/"C"_"V" = (4"R")/(3"R") = 4/3` = 1.33
Note that according to kinetic theory model of gases the specific heat capacity at constant volume and constant pressure are independent of temperature. But in reality it is not sure.
The specific heat capacity varies with the temperature.
Notes
Students should refer to the answer according to their questions.
संबंधित प्रश्न
A container has one mole of the monoatomic ideal gas. Each molecule has f degrees of freedom. What is the ratio of γ = `"C"_"p"/"C"_"v"`?
A sample of gas consists of μ1 moles of monoatomic molecules, μ2 moles of diatomic molecules and μ3 moles of linear triatomic molecules. The gas is kept at a high temperature. What is the total number of degrees of freedom?
Describe the total degrees of freedom for monoatomic molecule, diatomic molecule and triatomic molecule.
During an adiabatic process, the pressure of a mixture of monatomic and diatomic gases is found to be proportional to the cube of the temperature. Find the value of γ = `("C"_"P"/"C"_"V")`
A gas made of a mixture of 2 moles of oxygen and 4 moles of argon at temperature T. Calculate the energy of the gas in terms of RT. Neglect the vibrational modes.
A monoatomic gas molecule has ______.
Two monoatomic gas A and B occupying the same volume V are at the same temperature T and pressure P. If they are mixed, the resulting mixture has volume V and temperature T. The pressure of the mixture is:
Which of the following statement given below are correct for monoatomic gas?
I. Cv = `3/2 "R"`
II. Cp = `5/2 "R"`
III. Cp − Cv = 2R
IV. `"C"_"p"/"C"_"v" = 5/3`
A ballon has 5.0 g mole of helium at 7°C. Calculate
- the number of atoms of helium in the balloon
- the total internal energy of the system.
Calculate the number of degrees of freedom of molecules of hydrogen in 1 cc of hydrogen gas at NTP.
The average energy for molecules in one degree of freedom is ______.
Which statements are correct about degrees of freedom?
- A molecule with n degrees of freedom has n2 different ways of storing energy.
- Each degree of freedom is associated with `1/2`RT average energy per mole.
- A monoatomic gas molecule has 1 rotational degree of freedom whereas the diatomic molecule has 2 rotational degrees of freedom.
- CH4 has a total of 6 degrees of freedom.
Choose the correct answer from the option given below:
A gas has n degrees of freedom. The ratio of the specific heat of the gas at constant volume to the specific heat of the gas at constant pressure will be ______.
An ideal gas has molecules with 5 degrees of freedom. The ratio of specific heats at constant pressure (CP) and at constant volume (Cv) is ______.
