91影视

If a process is conducted rapidly,in an insulating chamber, then the process is known as adiabatic process. In the process, no exchange of heat can take place between the system and surrounding. Such a process must satisfy the relation-

${\mathbf{PV}}^{纬}\mathbf{=}\mathbf{constant}$

The average kinetic energy for each degree of freedom is equal to$\frac{1}{2}\mathbf{kT}$.

1. The volume of expanded gas is $V_1=3.00脳V_0$
2. The pressure after compression is$P_2={\left(3.00\right)}^{\frac{1}{3}}脳P_0$

The equation to be satisfied during the free expansion of an ideal gas is-

$P_i{V}_i=P_f{V}_f$

So, for the free expansion from the original state 0 to state 1, it becomes

$P_0{V}_0=P_1{V}_1$

For${\text{V}}_1=3.00脳{\text{V}}_0$; the equation becomes-

$\begin{array}{c}\frac{P_1}{P_0}=\frac{V_0}{V_1}\\ =\frac{V_0}{3.00脳V_0}\\ =\frac{1}{3}\end{array}$

The ratio of pressure in final state to initial state is$\frac{P_1}{P_0}=\frac{1}{3}$ .

For an adiabatic process, we have

$P{V}^{纬}=\text{constant}$

So, for this case, we can write

$P_1{V}_1^{纬}=P_2{V}_2^{纬}$

For, $V_1=3.00脳V_0$ and $P_2={\left(3.00\right)}^{\frac{1}{3}}脳P_0$, equation becomes-

$$\begin{gathered} \left(\frac{1}{3}{P}_0\right)脳{(3.00V_0)}^{纬}=(\left(3.00\right){}^{\frac{1}{3}}{P}_0){\left(V_0\right)}^{纬} \\ 纬鈭�1=\frac{1}{3} \\ 纬=\frac{4}{3} \end{gathered}$$

If f is the number of degree of freedom, then-

$\begin{array}{c}纬=1+\frac{2}{f}\\ =\frac{4}{3}\\ \begin{array}{c}\frac{4}{3}=1+\frac{2}{f}\\ f=6\end{array}\end{array}$

This shows that the gas is polyatomic.

We know that the average kinetic energy of an ideal gas is

$K_{avg}=\frac{3}{2}kT$

So,

$\begin{array}{c}\frac{K_{avg2}}{K_{avg1}}=\frac{T_2}{T_1}\\ =\frac{P_2}{P_1}\end{array}$

$$\begin{gathered} \frac{K_{avg2}}{K_{avg1}}=\frac{{\left(3.00\right)}^{\frac{1}{3}}{P}_0}{P_0}={3.00}^{\frac{1}{3}} \\ \frac{K_{avg2}}{K_{avg1}}=1.44 \end{gathered}$$

Ratio of the average kinetic energy per molecule is$\frac{K_{avg2}}{K_{avg1}}=1.44$