Collision Frequency Analysis

The problem requires calculating the collision frequency ($Z_A$) for gas A based on the given collision frequency for gas B ($Z_B$) and their respective molecular parameters.

We are provided with the following conditions for two ideal gases, A and B:

• Equal temperatures ($T_A = T_B$).
• Equal pressures ($P_A = P_B$).
• Equal number densities ($n_A = n_B$, derived from $P=nkT$).
• Molecular size relationship: $d_A = \frac{1}{2} d_B$.
• Molecular mass relationship: $m_A = 4 m_B$.
• Collision frequency in gas B: $Z_B = 32 \times 10^{18}$ /s.

Collision Frequency Formula Derivation

The collision frequency ($Z$) of a molecule in an ideal gas depends on the number density ($n$), molecular diameter ($d$), and average molecular speed ($\bar{v}$). The standard formula is:

$Z = \sqrt{2} \pi d^2 n \bar{v}$

The average molecular speed ($\bar{v}$) is related to temperature ($T$) and molecular mass ($m$) by:

$\bar{v} = \sqrt{\frac{8kT}{\pi m}}$

Substituting $\bar{v}$ into the collision frequency formula gives:

$Z = \sqrt{2} \pi d^2 n \sqrt{\frac{8kT}{\pi m}} = \frac{4 d^2 n \sqrt{kT}}{\sqrt{m}}$

Since $n$ and $T$ are constant for both gases, the collision frequency ($Z$) is proportional to the square of the molecular diameter ($d^2$) and inversely proportional to the square root of the molecular mass ($\sqrt{m}$):

$Z \propto \frac{d^2}{\sqrt{m}}$

Gas A Collision Frequency Calculation

We can find the ratio of the collision frequencies for gases A and B:

$\frac{Z_A}{Z_B} = \left(\frac{d_A}{d_B}\right)^2 \times \left(\frac{m_B}{m_A}\right)^{1/2}$

Using the given relationships $d_A = \frac{1}{2} d_B$ and $m_A = 4 m_B$:

• $\frac{d_A}{d_B} = \frac{1}{2}$
• $\frac{m_B}{m_A} = \frac{1}{4}$

Substitute these ratios into the formula:

$\frac{Z_A}{Z_B} = \left(\frac{1}{2}\right)^2 \times \left(\frac{1}{4}\right)^{1/2}$

$\frac{Z_A}{Z_B} = \frac{1}{4} \times \frac{1}{2} = \frac{1}{8}$

Now, calculate $Z_A$ using the known value of $Z_B$:

$Z_A = \frac{1}{8} \times Z_B = \frac{1}{8} \times (32 \times 10^{18} \text{ /s})$

$Z_A = 4 \times 10^{18} \text{ /s}$

Final Collision Frequency Value

The collision frequency in gas A is determined to be $4 \times 10^{18}$ /s.