Question

The solar corona is a very hot atmosphere surrounding the visible surface of the sun. X-ray emissions from the corona show that its temperature is about $2\times {10}^{6}K$. The gas pressure in the corona is about $0.03Pa$. Estimate the number density of particles in the solar corona.

Short answer

The number density of particles in the solar corona is$1.1\times {10}^{15}{m}^{-3}$.

Step-by-step solution

: Given information and Formula used 

Given : Temperature of X-ray emissions : $2\times {10}^{6}K$

The gas pressure is :$0.03Pa$

Theory used :
Number density is given by $numberdensity=\frac{N}{V}$
The ideal gas law is given by $PV=nRT$
we'll rewrite the equation of Ideal Gas Law for$V$in the form
$V=\frac{nRT}{P}$

Finding the number density

If we know the number of molecules $V$, we obtain the number of moles $n$by taking Avogadro's number$n=\frac{N}{N_A}$

Now put the expressions for $N\mathrm{and}V$in number density equation we get: $$\begin{gathered} numberdensity=\frac{n{N}_A}{{\displaystyle \frac{nRT}{p}}} \\ =\frac{N_{A}p}{RT} \end{gathered}$$
Now we put the values forrole="math" localid="1648406342009" $N_A,P,R,\mathrm{and}T$into this equation to get the number density

$$\begin{gathered} =\frac{(6.023x{10}^{23}\mathrm{molecules}/\mathrm{mol})(0.03Pa)}{(8.314J/molK)(2\times {10}^{6}K)} \\ =\mathbf{1}\mathbf{.}\mathbf{1}\mathbf{}\mathit{x}\mathbf{}{\mathbf{10}}^{\mathbf{15}}\mathbf{}{\mathit{m}}^{\mathbf{-}\mathbf{3}} \end{gathered}$$