One mole of an ideal gas of volume \(V\) and temperature \(T\) is allowed to expand adiabatically to volume \(2V\) while doing no external work. The universal gas constant is \(R\). What is the pressure of the gas after expansion?
Show Hint
Do not confuse "adiabatic free expansion" with "reversible adiabatic expansion".
In reversible adiabatic expansion, \(P V^{\gamma} = \text{constant}\).
But in free expansion, no work is done, temperature is constant, and the process is highly irreversible.
Simply apply the ideal gas law with \(T = \text{constant}\).
Step 1: Understanding the Question:
This question asks for the final pressure of an ideal gas undergoing free expansion.
The expansion is described as adiabatic, and no work is performed by the gas. Step 2: Key Formula or Approach:
• First Law of Thermodynamics:
\[ \Delta U = Q - W \]
• For an ideal gas, internal energy is a function of temperature only:
\[ U = n C_v T \]
• Ideal Gas Law:
\[ P V = n R T \]
Step 3: Detailed Explanation:
• The expansion process is adiabatic, meaning there is no heat exchange with the surroundings:
\[ Q = 0 \]
• The gas expands without doing any external work (free expansion into a vacuum):
\[ W = 0 \]
• Applying the First Law of Thermodynamics:
\[ \Delta U = 0 - 0 = 0 \]
• Since the internal energy of an ideal gas depends solely on its temperature, a constant internal energy (\(\Delta U = 0\)) means the temperature of the gas remains unchanged:
\[ T_{\text{final}} = T_{\text{initial}} = T \]
• The final volume of the gas is given as:
\[ V_{\text{final}} = 2V \]
• The number of moles is \(n = 1\).
• Applying the Ideal Gas Law to the final state:
\[ P_{\text{final}} \cdot V_{\text{final}} = n R T_{\text{final}} \]
\[ P_{\text{final}} \cdot (2V) = 1 \cdot R T \implies P_{\text{final}} = \frac{RT}{2V} \]
Step 4: Final Answer:
The final pressure of the gas after expansion is \(\frac{RT}{2V}\).
