Question 697: Thermodynamics & KTG

Question

Two cylinders A and B fitted with pistons contain equal amount of an ideal rigid diatomic gas at 303 K . The piston of cylinder $A$ is free to move and that of cylinder B is held fixed. The same amount heat is given to the gas in each cylinder. If the rise in temperature of the gas in cylinder B is 49 K , then the rise in temperature of the gas in $A$ is

Accepted Answer

Given:
Two cylinders, both contain equal amount of an ideal rigid diatomic gas at $303\,	ext{K}$.
Cylinder A: Piston is free (constant pressure process).
Cylinder B: Piston is fixed (constant volume process).
Equal amount of heat $Q$ is supplied to both.
In cylinder B (constant volume), the rise in temperature is $49\,	ext{K}$.
Find the rise in temperature of gas in A (constant pressure).
Step 1: Molar heat capacities
For a
rigid diatomic gas
(rotation possible, vibration neglected):
Degrees of freedom = $5$
$C_V = \frac{5}{2}R$
$C_P = C_V + R = \frac{7}{2}R$
Step 2: Heat supplied in each process
Let $n$ be the number of moles.
Cylinder B (Constant volume):
$ Q = n C_V \Delta T_B $
Given $\Delta T_B = 49\,	ext{K}$,
$ Q = n \left(\frac{5}{2}Right) 	imes 49 $
Cylinder A (Constant pressure):
$ Q = n C_P \Delta T_A $
But same $Q$ as above, so:
$ n C_P \Delta T_A = n C_V \Delta T_B $
Substitute $C_P = \frac{7}{2}R$, $C_V = \frac{5}{2}R$,
$ n \left(\frac{7}{2}Right) \Delta T_A = n \left(\frac{5}{2}Right) 	imes 49 $
Cancel $n$ and $\frac{1}{2}R$ from both sides:
$ 7 \Delta T_A = 5 	imes 49 $
$ \Delta T_A = \frac{5 	imes 49}{7} = \frac{245}{7} = 35\,	ext{K} $
Final Answer:
Option B: $35\,	ext{K}$

Answer

30 K

Answer

35 K

Answer

70 K

Answer

75 K

Thermodynamics & KTG

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