To analyze this scenario, we can refer to the Ideal Gas Law, which states:
\[ PV = nRT \]
Where:
- \( P \) = pressure,
- \( V \) = volume,
- \( n \) = number of moles of gas,
- \( R \) = ideal gas constant, and
- \( T \) = temperature (which is constant in this case).
Initially, the volume of the container is 10.0 nm. If the pressure fluctuates around a certain value \( P_1 \) at this volume, we can express the relationship as:
\[ P_1 \cdot V_1 = nRT \] \[ P_1 \cdot 10.0 = nRT \]
When the volume of the container is halved to 5.0 nm, we can denote the new pressure as \( P_2 \):
\[ P_2 \cdot V_2 = nRT \] \[ P_2 \cdot 5.0 = nRT \]
Since the amount of gas and temperature are constant, we know that \( nRT \) remains the same in both cases. Thus, we can set the equations equal to each other:
\[ P_1 \cdot 10.0 = P_2 \cdot 5.0 \]
Rearranging gives us:
\[ P_2 = \frac{P_1 \cdot 10.0}{5.0} \] \[ P_2 = 2 \cdot P_1 \]
This means that when the volume is halved, the pressure doubles. Therefore, if the initial pressure was \( P_1 \), the new pressure \( P_2 \) will be approximately double that initial value of \( P_1 \).
If you have a specific initial pressure \( P_1 \) value provided, you can substitute it in to obtain the exact value for \( P_2 \).
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