First slide

Kinetic theory of ideal gases

Question

A spherical balloon of volume V contains Helium gas at a pressure P. How many moles of Helium are there in the balloon if the average kinetic energy of the Helium atoms is $\bar{K}$ .

Moderate

A

$\frac{3 P V}{\bar{K} N_{A}}$
B

$\frac{P V}{\bar{3 K} N_{A}}$
C

$\frac{P V}{\bar{3 K}}$
D

$\frac{3 P V}{2 \bar{K} N_{A}}$

Solution

We know, $v_{R M S} = \sqrt{\frac{3 k T}{m}} = \sqrt{\frac{3 R T}{M}} = \sqrt{\frac{3 P V}{m a s s o f g a s}}$ , (1)
where m, M are mass of each molecule, molar mass of gas respectively.
$\Rightarrow \frac{1}{2} m v_{R M S}^{2} = \frac{3 k T}{2} = \bar{K}$
Also, from (1),
$\bar{K} = \frac{1}{2} m v_{R M S}^{2} = \frac{3 m P V}{2 (m a s s o f g a s)}$
But, $\frac{(m a s s o f g a s)}{m} = N$ (number of molecules).
$\Rightarrow \bar{K} = \frac{3 P V}{2 N} \Rightarrow N = \frac{3 P V}{2 \bar{K}}$
Now, number of moles = $n = \frac{N}{N_{A}} = \frac{3 P V}{2 \bar{K} N_{A}}$

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