The collision frequency ($Z$) of gas molecules is proportional to the square root of the absolute temperature ($T$) of the system. Mathematically, it can be represented as $Z \propto \sqrt{T}$. This implies that when the temperature changes, the collision frequency changes as well according to the formula:

$$Z_1 = Z_0 \sqrt{\frac{T_1}{T_0}}$$

where:

• $Z_1$ is the collision frequency at temperature $T_1$,
• $Z_0$ is the initial collision frequency at temperature $T_0$,
• $T_1$ is the final absolute temperature, and
• $T_0$ is the initial absolute temperature.

To solve the given problem, we first need to convert the provided temperatures from Celsius to Kelvin (since absolute temperature in Kelvin should be used):

• $T_0 = 27^{\circ}C = 27 + 273 = 300$ K
• $T_1 = 127^{\circ}C = 127 + 273 = 400$ K

Using the formula for collision frequency change and substituting the given values:

$$Z_1 = Z_0 \sqrt{\frac{400}{300}} = Z_0 \sqrt{\frac{4}{3}}$$

This can be simplified to:

$$Z_1 = Z_0 \times \frac{2}{\sqrt{3}}$$

Thus, the collision frequency of the system at $127^{\circ}C$ is $\frac{2}{\sqrt{3}}$ times the collision frequency at $27^{\circ}C$, making Option B the correct answer:

$$\frac{2}{\sqrt{3}} \mathrm{Z}$$