Question 966: Thermodynamics & KTG

Question

Two gases $A$ and $B$ are filled at the same pressure in separate cylinders with movable pistons of radius $r_A$ and $r_B$, respectively. On supplying an equal amount of heat to both the systems reversibly under constant pressure, the pistons of gas $A$ and $B$ are displaced by 16 cm and 9 cm , respectively. If the change in their internal energy is the same, then the ratio $\frac{r_A}{r_B}$ is equal to

Accepted Answer

The problem involves two gases, $A$ and $B$, at the same pressure in separate cylinders with movable pistons of radii $r_A$ and $r_B$. Equal amounts of heat are supplied to both cylinders reversibly under constant pressure, displacing the pistons by 16 cm for gas $A$ and 9 cm for gas $B$. Given that the change in their internal energy is the same, we need to find the ratio $\frac{r_A}{r_B}$.
First, we apply the first law of thermodynamics:
$ \Delta Q = \Delta U + P \Delta V $
Since $\Delta Q$ and $\Delta U$ are the same for both gases, their work done $W_A$ and $W_B$ is also equal. Therefore:
$ (P \Delta V)_A = (P \Delta V)_B $
With constant pressure $P$, the relationship becomes:
$ A_A d_A = A_B d_B $
where $A$ is the area of the piston. We know:
$ \pi r_A^2 d_A = \pi r_B^2 d_B $
Simplifying, the ratio of the radii is:
$ \frac{r_A}{r_B} = \left(\frac{d_B}{d_A}\right)^{\frac{1}{2}} = \left(\frac{9}{16}\right)^{\frac{1}{2}} $
Calculating the square root gives:
$ \frac{r_A}{r_B} = \frac{3}{4} $

Answer

$\frac{2}{\sqrt{3}}$

Answer

$\frac{\sqrt{3}}{2}$

Answer

$\frac{4}{3}$

Answer

$\frac{3}{4}$

Thermodynamics & KTG

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