The number density of molecules of a gas depends on their distance from the origin as, . Then the total number of molecules is proportional to:

Question

The number density of molecules of a gas depends on their distance $' r'$ from the origin as, $n (r) = n_{0} e^{- α r^{4}}$ . Then the total number of molecules is proportional to:

Smart Achievers · Accepted Answer

(1)

$N = \int ρ (d v) = \int_{0}^{r} n_{0} e^{- a r^{4}} \times 4 π r^{2} d r = 4 π n_{0} \int_{0}^{r} r^{2} (e^{- a r^{4}}) d r \propto n_{0} a^{- 3 / 4}$

Solution

Q.
The number density of molecules of a gas depends on their distance $' r'$ from the origin as, $n (r) = n_{0} e^{- α r^{4}}$ . Then the total number of molecules is proportional to:

1 $n_{0} α^{- 3 / 4}$

2 $\sqrt{n_{0}} α^{1 / 2}$

3 $n_{0} α^{1 / 4}$

4 $n_{0} α^{- 3}$

Ans.
(1)

$N = \int ρ (d v) = \int_{0}^{r} n_{0} e^{- a r^{4}} \times 4 π r^{2} d r = 4 π n_{0} \int_{0}^{r} r^{2} (e^{- a r^{4}}) d r \propto n_{0} a^{- 3 / 4}$

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Solution

Q. The number density of molecules of a gas depends on their distance 'r' from the origin as, n(r)=n0e-αr4. Then the total number of molecules is proportional to:

1 n0α-3/4

2 n0α1/2

3 n0α1/4

4 n0α-3

Ans. (1) N=∫ρ(dv)=∫0rn0e-ar4×4πr2dr=4πn0∫0rr2(e-ar4)dr∝n0a-3/4