Question:
An ideal gas is expanding such that
\[
P^2T=\text{constant}.
\]
The coefficient of volume expansion of the gas is
\[
(P=\text{Pressure},\; T=\text{Temperature})
\]
An ideal gas is expanding such that
\[
P^2T=\text{constant}.
\]
The coefficient of volume expansion of the gas is
\[
(P=\text{Pressure},\; T=\text{Temperature})
\]
Show Hint
If
\[
V\propto T^n,
\]
then
\[
\alpha
=
\frac{1}{V}\frac{dV}{dT}
=
\frac{n}{T}.
\]
Here,
\[
V\propto T^{3/2},
\]
hence
\[
\alpha=\frac{3}{2T}.
\]
Updated On: Jul 9, 2026
- \(\dfrac{1}{T}\)
- \(\dfrac{2}{T}\)
- \(\dfrac{3}{T}\)
- \(\dfrac{3}{2T}\) \bigskip
Show Solution
Verified By Collegedunia
The Correct Option is D
Solution and Explanation
Concept:
The coefficient of volume expansion is defined as
\[
\alpha
=
\frac{1}{V}
\left(\frac{\partial V}{\partial T}\right).
\]
For an ideal gas,
\[
PV=nRT.
\]
Step 1: Use the given condition. Given, \[ P^2T=\text{constant}. \] Therefore, \[ P^2=\frac{k}{T}, \] where \(k\) is a constant. Hence, \[ P=\frac{\sqrt{k}}{\sqrt{T}} \propto T^{-1/2}. \]
Step 2: Express volume in terms of temperature. Using the ideal gas equation, \[ V=\frac{nRT}{P}. \] Since \[ P\propto T^{-1/2}, \] we get \[ V\propto T\times T^{1/2}. \] \[ V\propto T^{3/2}. \] Let \[ V=C\,T^{3/2}, \] where \(C\) is a constant.
Step 3: Differentiate with respect to temperature. \[ \frac{dV}{dT} = \frac{3}{2}CT^{1/2}. \] Therefore, \[ \alpha = \frac{1}{CT^{3/2}} \left(\frac{3}{2}CT^{1/2}\right). \] \[ \alpha = \frac{3}{2T}. \]
Step 4: Write the final answer. \[ \boxed{\alpha=\frac{3}{2T}} \] \[ \boxed{\text{Answer = (D)}} \]
Step 1: Use the given condition. Given, \[ P^2T=\text{constant}. \] Therefore, \[ P^2=\frac{k}{T}, \] where \(k\) is a constant. Hence, \[ P=\frac{\sqrt{k}}{\sqrt{T}} \propto T^{-1/2}. \]
Step 2: Express volume in terms of temperature. Using the ideal gas equation, \[ V=\frac{nRT}{P}. \] Since \[ P\propto T^{-1/2}, \] we get \[ V\propto T\times T^{1/2}. \] \[ V\propto T^{3/2}. \] Let \[ V=C\,T^{3/2}, \] where \(C\) is a constant.
Step 3: Differentiate with respect to temperature. \[ \frac{dV}{dT} = \frac{3}{2}CT^{1/2}. \] Therefore, \[ \alpha = \frac{1}{CT^{3/2}} \left(\frac{3}{2}CT^{1/2}\right). \] \[ \alpha = \frac{3}{2T}. \]
Step 4: Write the final answer. \[ \boxed{\alpha=\frac{3}{2T}} \] \[ \boxed{\text{Answer = (D)}} \]
Was this answer helpful?
0
0
Top AP EAPCET Physics Questions
- A body is projected from the ground at an angle of $\tan^{-1} \left( \frac{8}{7}\right)$ with the horizontal. The ratio of the maximum height attained by it to its range is
- A lens forms real and virtual images of an object, when the object is at $u_{1}$ and $u_{2}$ distances respectively. If the size of the virtual image is double that of the real image, then the focal length of the lens is (take, the magnification of the real image as $m$ )
- Two spheres $P$ and $Q$, each of mass $200\, g$ are attached to a string of length one metre as shown in the figure. The string and the spheres are then whirled in a horizontal circle about $O$ at a constant angular speed. The ratio of the tension in the string between $P$ and $Q$ to that of between $P$ and $O$ is $(P$ is at mid-point of the line joining $O$ and $Q$ )
- The radii of the two columns in a U - tube are r1 and r2. When a liquid of density p (angle of contact is $ 0{}^\circ $ ) is filled in it, the level difference of the liquid in the two arms is h. The surface tension of the liquid is: (g = acceleration due to gravity)
- A rigid metallic sphere is spinning around its own axis in the absence of external torque. If the temperature is raised, its volume increases by $9 \%$. The change in its angular speed is
View More Questions
Top AP EAPCET Gas laws Questions
- In an isothermal process, which of the following remains constant?
- The temperature of a given mass of an ideal gas is changed from 27^ to 327^ at constant pressure. If the initial volume of the gas is V, then its final volume will be:
- In which of the following processes does the internal energy of the gas remain unaffected?
Top AP EAPCET Questions
- A body is projected from the ground at an angle of $\tan^{-1} \left( \frac{8}{7}\right)$ with the horizontal. The ratio of the maximum height attained by it to its range is
- The four quantum numbers of the valence electron of potassium are :
- A lens forms real and virtual images of an object, when the object is at $u_{1}$ and $u_{2}$ distances respectively. If the size of the virtual image is double that of the real image, then the focal length of the lens is (take, the magnification of the real image as $m$ )
- Two spheres $P$ and $Q$, each of mass $200\, g$ are attached to a string of length one metre as shown in the figure. The string and the spheres are then whirled in a horizontal circle about $O$ at a constant angular speed. The ratio of the tension in the string between $P$ and $Q$ to that of between $P$ and $O$ is $(P$ is at mid-point of the line joining $O$ and $Q$ )
- The volume of $0.02\, M$ acidified permanganate solution required for complete reaction of $60\, mL$ of $0.01\, MI^{-}$ ion solution to form $I_2$ in $mL$ is
View More Questions