1. Use the relation for mean free time

The mean time between successive collisions is

$au = \frac{\lambda}{\bar v}$

where:

• $\lambda$ = mean free path
• $\bar v$ = average molecular speed

2. Find how mean free path depends on $T$ at constant pressure

For a dilute gas,

$\lambda = \frac{1}{\sqrt{2}\,n\sigma}$

where $n$ is number density and $\sigma$ is collision cross-section.

Using the ideal gas law,

$P = nkT \implies n = \frac{P}{kT}$

Since $P$ is constant,

$n \propto \frac{1}{T}$

Hence,

$\lambda \propto \frac{1}{n} \propto T$

3. Find how average speed depends on $T$

For gas molecules,

$\bar v \propto \sqrt{T}$

4. Now find the dependence of $\tau$ on $T$

$\tau = \frac{\lambda}{\bar v} \propto \frac{T}{\sqrt{T}} = \sqrt{T}$

So the mean time between collisions varies as

$\boxed{\sqrt{T}}$

5. Check options

• A: $\sqrt{T}$ ✅
• B: $T$ ❌
• C: $\dfrac{1}{T}$ ❌
• D: $\dfrac{1}{\sqrt{T}}$ ❌

Therefore, the correct option is

$\boxed{A}$

6. Comparison with stored answer

Stored correct answer: $D$

My derived answer is $A$. The stored answer appears incorrect because at constant pressure, number density decreases as $1/T$, making mean free path increase as $T$, while molecular speed increases only as $\sqrt{T}$. Therefore the collision time increases as $\sqrt{T}$, not decreases as $1/\sqrt{T}$.