Question

Interstellar space (far from any stars) contains atomic hydrogen (H) with a density of 1 atom/cm3 and at a temperature of about 2.7 K. Determine (a) the pressure in interstellar space, (b) root-mean square speed of the atoms and (c) The kinetic energy stored in 1 km3 of space.

Answer 1

Explanation:

Given that,

Number density $n= 1\ atom/cm^{3} =10^{6}\ atom/m^3$

Temperature = 2.7 K

(a). We need to calculate the pressure in interstellar space

Using ideal gas equation

$PV=nRT$

$P=\dfrac{nRT}{V}$

$P=\dfrac{10^{6}\times8.314\times2.7}{6.023\times10^{23}}$

$P=3.727\times10^{-17}\ Pa$

$P=36.78\times10^{-23}\ atm$

The pressure in interstellar space is $36.78\times10^{-23}\ atm$

(b). We need to calculate the root-mean square speed of the atom

Using formula of rms

$v_{rms}=\sqrt{\dfrac{3RT}{Nm}}$

Put the value into the formula

$v_{rms}=\sqrt{\dfrac{3\times8.314\times2.7}{1.007\times10^{-3}}}$

$v_{rms}=258.6\ m/s$

The root-mean square speed of the atom is 258.6 m/s.

(c). We need to calculate the  kinetic energy

Average kinetic energy of atom

$E=\dfrac{3}{2}kT$

Where, k = Boltzmann constant

Put the value into the formula

$E=\dfrac{3}{2}\times1.38\times10^{-23}\times2.7$

$E=5.58\times10^{-23}\ J$

The kinetic energy stored in 1 km³ of space is $5.58\times10^{-23}\ J$.

Hence, This is the required solution.