Question:medium
An ideal gas undergoes a process maintaining relation between pressure (\(P\)) and volume (\(V\)) as \(P = P_o \left(1 + \left(\frac{V_o}{V}\right)^2\right)^{-1}\), where \(P_o\) and \(V_o\) are constants. If two samples A and B (two moles each) with initial volumes \(V_o\) and \(3V_o\) respectively undergo above mentioned process and attain same pressure, then the difference at the temperatures of these samples, \(T_B - T_A\) is ______. (\(R\) = gas constant)
An ideal gas undergoes a process maintaining relation between pressure (\(P\)) and volume (\(V\)) as \(P = P_o \left(1 + \left(\frac{V_o}{V}\right)^2\right)^{-1}\), where \(P_o\) and \(V_o\) are constants. If two samples A and B (two moles each) with initial volumes \(V_o\) and \(3V_o\) respectively undergo above mentioned process and attain same pressure, then the difference at the temperatures of these samples, \(T_B - T_A\) is ______. (\(R\) = gas constant)
Updated On: Jun 6, 2026
- \(\frac{9P_0V_0}{8R}\)
- \(\frac{11P_0V_0}{10R}\)
- \(\frac{7P_0V_0}{6R}\)
- \(\frac{13P_0V_0}{11R}\)
Show Solution
The Correct Option is B
Solution and Explanation
Step 1: Understanding the Concept:
The phrasing refers to the difference in temperatures of the two specified macro-states \(A\) and \(B\) on the given thermodynamic process curve. We simply need to find the pressure at the given volumes using the provided $P$-$V$ relation and then calculate the temperature using the ideal gas law.
Step 2: Key Formula or Approach:
The pressure-volume relationship is \(P = \frac{P_0}{1 + (V_0/V)^2}\).
The Ideal Gas Law is \(PV = nRT\), where \(n = 2\).
Therefore, temperature is given by \(T = \frac{PV}{nR} = \frac{PV}{2R}\).
Step 3: Detailed Explanation:
Let's analyze state \(A\) with volume \(V_A = V_0\):
\(P_A = P_0 \left[ 1 + \left(\frac{V_0}{V_0}\right)^2 \right]^{-1} = P_0 [1 + 1^2]^{-1} = \frac{P_0}{2}\).
Temperature \(T_A = \frac{P_A V_A}{2R} = \frac{(\frac{P_0}{2}) V_0}{2R} = \frac{P_0 V_0}{4R}\).
Let's analyze state \(B\) with volume \(V_B = 3V_0\):
\(P_B = P_0 \left[ 1 + \left(\frac{V_0}{3V_0}\right)^2 \right]^{-1} = P_0 \left[ 1 + \frac{1}{9} \right]^{-1} = P_0 \left[ \frac{10}{9} \right]^{-1} = \frac{9P_0}{10}\).
Temperature \(T_B = \frac{P_B V_B}{2R} = \frac{(\frac{9P_0}{10}) (3V_0)}{2R} = \frac{27 P_0 V_0}{20R}\).
Now calculate the difference \(T_B - T_A\):
\(T_B - T_A = \frac{27 P_0 V_0}{20R} - \frac{P_0 V_0}{4R}\)
Find a common denominator (20):
\(T_B - T_A = \frac{27 P_0 V_0 - 5 P_0 V_0}{20R} = \frac{22 P_0 V_0}{20R} = \frac{11 P_0 V_0}{10R}\).
Step 4: Final Answer:
The difference in temperatures is \(\frac{11P_0 V_0}{10R}\).
The phrasing refers to the difference in temperatures of the two specified macro-states \(A\) and \(B\) on the given thermodynamic process curve. We simply need to find the pressure at the given volumes using the provided $P$-$V$ relation and then calculate the temperature using the ideal gas law.
Step 2: Key Formula or Approach:
The pressure-volume relationship is \(P = \frac{P_0}{1 + (V_0/V)^2}\).
The Ideal Gas Law is \(PV = nRT\), where \(n = 2\).
Therefore, temperature is given by \(T = \frac{PV}{nR} = \frac{PV}{2R}\).
Step 3: Detailed Explanation:
Let's analyze state \(A\) with volume \(V_A = V_0\):
\(P_A = P_0 \left[ 1 + \left(\frac{V_0}{V_0}\right)^2 \right]^{-1} = P_0 [1 + 1^2]^{-1} = \frac{P_0}{2}\).
Temperature \(T_A = \frac{P_A V_A}{2R} = \frac{(\frac{P_0}{2}) V_0}{2R} = \frac{P_0 V_0}{4R}\).
Let's analyze state \(B\) with volume \(V_B = 3V_0\):
\(P_B = P_0 \left[ 1 + \left(\frac{V_0}{3V_0}\right)^2 \right]^{-1} = P_0 \left[ 1 + \frac{1}{9} \right]^{-1} = P_0 \left[ \frac{10}{9} \right]^{-1} = \frac{9P_0}{10}\).
Temperature \(T_B = \frac{P_B V_B}{2R} = \frac{(\frac{9P_0}{10}) (3V_0)}{2R} = \frac{27 P_0 V_0}{20R}\).
Now calculate the difference \(T_B - T_A\):
\(T_B - T_A = \frac{27 P_0 V_0}{20R} - \frac{P_0 V_0}{4R}\)
Find a common denominator (20):
\(T_B - T_A = \frac{27 P_0 V_0 - 5 P_0 V_0}{20R} = \frac{22 P_0 V_0}{20R} = \frac{11 P_0 V_0}{10R}\).
Step 4: Final Answer:
The difference in temperatures is \(\frac{11P_0 V_0}{10R}\).
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