Question

Nitrogen gas is filled in an insulated container. If $\alpha$ fraction of moles dissociates without exchange of any energy, then the fractional change in its temperature is

Answer 1

A formula of internal energy at temperature T and no of a mole of gas n is $n{C}_{v}T$ .

where Cv is molar heat capacity at constant volume.

N₂ is diatomic molecule so, $Cv=5R/2$

Let initial temperature is T₁ and number of mole of N₂ is n

Then, internal energy $E=n\times 5R/2\times T₁$

$\Rightarrow T₁=2E/5nR-------\left(1\right)$

after dissociation of N₂ gas ;

$N₂---------->2N$

at t = 0

number of mole of $N₂=n$

number of moles of $N=0$

After dissociation by α fractional ,

number of moles of $N₂gas=n(1-\alpha )$

number of moles of $N=2n\alpha$

Now, Let temperature be T₂ after dissociation ,

Then, internal energy $E=energyofN₂+energyofN$

$E=n(1-\alpha )\times 5R/2\times T₂+2n\alpha \times 3R/2\times T₂[Nismonoatomicso,Cv=3R/2]$

$E=T₂nR/2\left[5\right(1-\alpha )+6\alpha ]=nRT₂[5+\alpha ]/2$

$T₂=2E/nR[5+\alpha ]------\left(2\right)$

Now, fractional change in temperature is $\frac{T_2-T_1}{T_1}$

Fraction change $=\frac{2E}{nR}5+\alpha -\frac{2E}{5nR}$

$=1/(5+\alpha )-1/5/1/5$

$=-\alpha /(5+\alpha )$

Hence, fractional change in temperature is $-\alpha /(5+\alpha )$