Question 747: Thermodynamics & KTG

Question

Two cylinders A and B fitted with piston contain equal amount of an ideal diatomic as at temperature ' T ' K . The piston of cylinder A is free to move while that of B is held fixed. The same amount of heat is given to the gas in each cylinder. If the rise temperature of the gas in A is ' $\mathrm{dT}_{\mathrm{A}}$ ', then the rise in temperature of the gas in cylinder B is $\left(\gamma=\frac{\mathrm{C}_{\mathrm{p}}}{\mathrm{C}_{\mathrm{v}}}\right)$

Accepted Answer

When heat is applied to both cylinders A and B, they behave differently due to the piston's conditions. In cylinder A, the piston is free to move, so the heat is applied at constant pressure. Meanwhile, in cylinder B, the piston is fixed, meaning heat is applied at constant volume.
To find the relationship between the rise in temperature for cylinders A and B, note that the amount of heat added to both cylinders is the same:
$ Q_A = Q_B $
For cylinder A (constant pressure), the heat added is given by:
$ Q_A = nC_p \, dT_A $
For cylinder B (constant volume), the heat added is:
$ Q_B = nC_v \, dT_B $
Since $Q_A = Q_B$, equate the two expressions:
$ nC_p \, dT_A = nC_v \, dT_B $
Thus, the expression for the temperature rise in cylinder B becomes:
$ dT_B = \frac{C_p}{C_v} \, dT_A = \gamma \, dT_A $
Therefore, the increase in temperature of the gas in cylinder B is $\gamma \, dT_A$, where $\gamma$ is the heat capacity ratio $\frac{C_p}{C_v}$.

Answer

$2 \mathrm{dt}_{\mathrm{A}}$

Answer

$\frac{\mathrm{dT}_{\mathrm{A}}}{2}$

Answer

$\mathrm{\gamma d T_A}$

Answer

$\frac{\mathrm{dT}_A}{\gamma}$

Thermodynamics & KTG

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