91影视

Formula for quantum volume is:

${谓}_Q={\left(\frac{h}{\sqrt{2\mathrm{蟺尘办罢}}}\right)}^3$

where,$T$is temperature,role="math" localid="1647248715772" $m$is molecule mass,$h$is Planck's constant,$k$is Boltzmann constant.

Mass of $N_2$ is calculated as:

Substitute variables in quantum volume formula:

$$\begin{gathered} {谓}_Q={\left(\frac{6.63脳{10}^{-34}J路s}{\sqrt{2蟺(46.48脳{10}^{-27}kg)(1.38脳{10}^{-23}J/K)(300K)}}\right)}^3 \\ ={\left(\frac{6.63脳{10}^{-34}}{\sqrt{347.91脳{10}^{-50}}}\right)}^3 \\ =6.9脳{10}^{-33}{m}^3 \end{gathered}$$

So, quantum volume of$N_2$molecule is$6.9脳{10}^{-33}{m}^3$.

Ideal gas equation is $PV=NkT$

So, $\frac{V}{N}=\frac{kT}{P}$

Substitute Pressure,$P=1.013脳{10}^{5}N/m^2$, Temperature,$T=300K$.

role="math" localid="1647253225593" $$\begin{gathered} \frac{V}{N}=\frac{(1.38脳{10}^{-23}J/K)(300K)}{1.013脳{10}^{5}N/m^2} \\ =4脳{10}^{-26}{m}^3 \end{gathered}$$

In quantum statistics, $\frac{V}{N}鈮�{谓}_Q$

$$\begin{gathered} \frac{kT}{P}鈮�{\left(\frac{h}{\sqrt{2\mathrm{蟺尘办罢}}}\right)}^3 \\ T.T^{\frac{3}{2}}鈮�\frac{h^3}{{\left(2\mathrm{蟺尘碍}\right)}^{{\displaystyle \frac{3}{2}}}}脳\frac{P}{k} \\ {T}^{\frac{5}{2}}鈮�\frac{P{h}^3}{{\left(2\mathrm{蟺尘}\right)}^{{\displaystyle \frac{3}{2}}}{k}^{{\displaystyle \frac{5}{2}}}} \\ T鈮�\frac{1}{k}{\left[\frac{P^2{h}^6}{{\left(2\mathrm{蟺尘}\right)}^3}\right]}^{\frac{1}{5}} \end{gathered}$$

Substitute the values of variables in above equation:

$$\begin{gathered} T鈮�\frac{1}{1.38脳{10}^{-23}J/K}{\left[\frac{{(1.013脳{10}^{5}N/m^2)}^2{(6.63脳{10}^{-34}J路s)}^6}{{\left(2蟺\right(28脳1.67脳{10}^{-27}kg\left)\right)}^3}\right]}^{\frac{1}{5}} \\ T=5.681K \end{gathered}$$

So, temperature of the system is$5.681K$.