Question

if the temperature of a diatomic ideal gas initialy at T and cooled to 1/4T calculate
1-the ratio of root mean square velocities and ratio of mean kinetic energies in two states
2-Cp,Cv and gama constant of the gas

Answer 1

Answer on Question #43050-Physics-Molecular Physics-Thermodynamics

If the temperature of a diatomic ideal gas initially at T and cooled to 1/4T calculate

1- the ratio of root mean square velocities and ratio of mean kinetic energies in two states

2- Cp, Cv and gamma constant of the gas

Solution

1. The root mean square velocity is

$$v _ {r m s} = \sqrt {\frac {3 k T}{m}}.$$

The ratio of root mean square velocities in two states is

$$\frac {v _ {2 r m s}}{v _ {1 r m s}} = \frac {\sqrt {\frac {3 k T _ {2}}{m}}}{\sqrt {\frac {3 k T _ {1}}{m}}} = \sqrt {\frac {T _ {2}}{T _ {1}}} = \sqrt {\frac {1}{4} T} = \frac {1}{2}.$$

The mean kinetic energy of molecule is $\frac{3}{2} kT$ .

The ratio of mean kinetic energies in two states

$$\frac {\frac {3}{2} k T _ {2}}{\frac {3}{2} k T _ {1}} = \frac {T _ {2}}{T _ {1}} = \frac {\frac {1}{4} T}{T} = \frac {1}{4}.$$

2. We have a diatomic ideal gas. Its molar heat capacities at constant volume is

$$c _ {V} = \frac {5}{2} R = 20.8 \frac {J}{molK}.$$

Its molar heat capacities at constant pressure is

$$c _ {p} = c _ {V} + R = \frac {7}{2} R = 29.1 \frac {J}{molK}.$$

Its gamma constant is

$$\gamma = \frac {c _ {p}}{c _ {V}} = \frac {\frac {7}{2} R}{\frac {5}{2} R} = \frac {7}{5} = 1.4.$$

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