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The problem involves an ideal gas undergoing an adiabatic process. In such processes, the relationship between pressure (\( P \)) and volume (\( V \)) is governed by the adiabatic condition:

\[
P_1 V_1^\gamma = P_2 V_2^\gamma
\]

where \( P_1 \) and \( V_1 \) are the initial pressure and volume, and \( P_2 \) and \( V_2 \) are the final pressure and volume. \( \gamma \) (gamma) is the adiabatic index or the ratio of specific heats \( C_p / C_v \), which depends on the type of gas (typically \( \gamma \) = 1.4 for diatomic gases like air).

To calculate the ratio \( \frac{P_1}{P_2} \), we rearrange the adiabatic condition:

\[
\frac{P_1}{P_2} = \left( \frac{V_2}{V_1} \right)^\gamma
\]

### Given values:
- Initial volume \( V_1 = 3.7 \times 10^3 \, \text{cm}^3 \)
- Final volume \( V_2 = 8.2 \times 10^3 \, \text{cm}^3 \)

Let me calculate the ratio of the initial pressure to the final pressure.

I'll now compute the result for \( \gamma = 1.4 \).

An ideal gas with a volume of 3.7 × 10^3 cm^3 expands adiabatically and its volume becomes 8.2 × 10^3 cm^3. Calculate the ratio of initial pressure of the gas to final pressure of the gas. Give your answer to an appropriate number of significant figures.

Learn how to calculate the pressure ratio during the adiabatic expansion of an ideal gas, using the given volumes and applying the adiabatic process formula. Suitable for high school and undergraduate physics students.

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