The initial pressure and volume of an ideal gas are \( P_0 \) and \( V_0 \). The final pressure of the gas when the gas is suddenly compressed to volume \( V_0/4 \) will be: (Given \( \gamma \) = ratio of specific heats at constant pressure and at constant volume)
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In thermodynamic problems, words like "sudden" or "rapid" imply an adiabatic process (\( PV^\gamma = \text{const} \)), while "slow" implies an isothermal process (\( PV = \text{const} \)). \
Step 1: Understanding the Concept:
When a gas is "suddenly" compressed or expanded, there is no time for heat exchange between the system and its surroundings. Such a process is considered an adiabatic process. In an adiabatic process, the pressure and volume follow a specific power-law relationship. Step 2: Key Formula or Approach:
For an adiabatic process:
\[ PV^{\gamma} = \text{constant} \]
Or, \( P_1 V_1^{\gamma} = P_2 V_2^{\gamma} \). Step 3: Detailed Explanation:
1. Identify the initial states: \( P_1 = P_0 \) and \( V_1 = V_0 \).
2. Identify the final states: \( P_2 = ? \) and \( V_2 = \frac{V_0}{4} \).
3. Substitute these into the adiabatic equation:
\[ P_0 (V_0)^{\gamma} = P_2 \left( \frac{V_0}{4} \right)^{\gamma} \]
4. Solve for \( P_2 \):
\[ P_2 = \frac{P_0 (V_0)^{\gamma}}{(\frac{V_0}{4})^{\gamma}} \]
\[ P_2 = P_0 \left( \frac{V_0}{V_0/4} \right)^{\gamma} \]
\[ P_2 = P_0 (4)^{\gamma} \] Step 4: Final Answer
The final pressure of the gas is \( P_0(4)^{\gamma} \).