An ideal gas goes through a reversible cycle \$a\rightarrow b\rightarrow c\rightarrow d\$ has the V- T diagram shown below. Process \$d\rightarrow a \$ and \$b\rightarrow c\$ are adiabatic.The corresponding P-V diagram for the process is ( all figure are schematic and not drawn to scale).

In process \$a \rightarrow b\$ and \$c \rightarrow d\$ the volume is directly proportional to temperature of gas, hence the process is isobaric, however, the slope of curve is different in each case. using ideal gas equation\$PV=nRT \Rightarrow V=\dfrac{nR}{P} T\$ slope \$m=\dfrac{nR}{P}\$ for a high pressure the slope of the V-T curve will be less. In given diagram the slope of \$c \rightarrow d\$ process is lower than \$a \rightarrow b\$ process, hence process \$a \rightarrow b\$ is at higher pressure, from which option B and D can be ruled out.Now initial rate of change of volume w.r.t. temperature is very less from \$b \rightarrow c\$ and \$d \rightarrow a\$ as shown in V-T diagram.hence slope of P-V curve should tend to \$\lim_{\Delta V\rightarrow 0}\dfrac{\partial P }{\partial v}=\infty\$ as initial change in volume is nearly zero, which can be verified in option AHence correct answer is option A.

A compressive force, F is applied at the two ends of a long thin steel rod. It is hated, simultaneously, such that its temperature increases by \$\triangle T \$ . The net change in its length is zero. Let l be the length of the rod, A its area of cross-section, Y its Young,s modulus, and \$\alpha\$ its coefficient of linear expansion. Then, F is equal to :

\$\dfrac{\Delta L}{L} = \alpha \Delta T\$\$\dfrac{Stress}{Strain}= Y\$\$\dfrac{F}{A\alpha \Delta T} = Y\$\$\dfrac{F}{A} = Y \alpha \Delta T\$\$F= A Y \alpha \Delta T\$

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Thermal Properties Of Matter

Q: A leak proof cylinder of length \$1m\$, made of a metal which has low coefficient of expansion is floating vertically in water at \$0^oC\$ such that its height above the water surface is \$20cm\$. When the temperature of water is increased to \$4^oC\$, the height of the cylinder above the water surface becomes \$21cm\$. The density of water at \$T = 4^oC\$, relative to the density at \$T = 0^oC\$ is close to:

Let \$A\$ be the area of cross section of the cylinder.\$p_0\$ be the density of water at \$0^oC\$\$p_4\$ be the density of water at \$4^oC\$.Total length of the cylinder is \$100cm\$Given that the length of the cylinder above the surface of water is \$20cm\$At \$0^oC\$, the weight of the cylinder is balanced by the buoyant force.\$p_0A \times 0.8 \times g = w\$ [Buoyant force \$= pA lg\$] ...(i)At 4^oC\$, thef weight of the cylinder is balanced by the buoyant force.\$p_4A\times 0.79\times g = w\$ ...(ii)Dividing (i) by (ii)\$\dfrac{p_0\times 0.8}{p_4 \times 0.79} = 1\$, \$p_4 = \dfrac{p_0\times 0.8}{0.79} = 0.012p_0\$\$\dfrac{p_4}{p_0} = 1.012 \simeq 1.01

Q: Which of the following shows the correct relationship between the pressure \$'P'\$ and density \$\rho\$ of an ideal gas at constant temperature ?

From ideal gas law:\$PV=nRT\$\$P=\cfrac{MnRT}{MV}, \quad M:\$ Molecular mass number But \$\rho=Mn/V\$\$P=\rho RT/M\$At constant temperature, \$P \propto \rho\$Hence, the correct graph is (d).

Physics
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An ideal gas goes through a reversible cycle $a \to b \to c \to d$ has the V- T diagram shown below. Process $d \to a$ and $b \to c$ are adiabatic.
The corresponding P-V diagram for the process is ( all figure are schematic and not drawn to scale).

Hard

JEE Mains

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A leak proof cylinder of length $1 m$ , made of a metal which has low coefficient of expansion is floating vertically in water at $0^{o} C$ such that its height above the water surface is $20 c m$ . When the temperature of water is increased to $4^{o} C$ , the height of the cylinder above the water surface becomes $21 c m$ . The density of water at $T = 4^{o} C$ , relative to the density at $T = 0^{o} C$ is close to:

1.26

1.03

1.04

1.01

Medium

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A compressive force, F is applied at the two ends of a long thin steel rod. It is hated, simultaneously, such that its temperature increases by $△ T$ . The net change in its length is zero. Let l be the length of the rod, A its area of cross-section, Y its Young,s modulus, and $α$ its coefficient of linear expansion. Then, F is equal to :

l^{2} Y α △ T

l A Y α △ T

\frac{A Y}{α △ T}

A Y α △ T

Medium

JEE Mains

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Which of the following shows the correct relationship between the pressure $^{'} P^{'}$ and density $ρ$ of an ideal gas at constant temperature ?

Medium

JEE Mains

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Thermal Properties Of Matter

Physics 6338 Views

An ideal gas goes through a reversible cycle a→b→c→d has the V- T diagram shown below. Process d→a and b→c are adiabatic. The corresponding P-V diagram for the process is ( all figure are schematic and not drawn to scale).

Which of the following shows the correct relationship between the pressure ′P′ and density ρ of an ideal gas at constant temperature ?

Physics
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An ideal gas goes through a reversible cycle $a \to b \to c \to d$ has the V- T diagram shown below. Process $d \to a$ and $b \to c$ are adiabatic.
The corresponding P-V diagram for the process is ( all figure are schematic and not drawn to scale).

Which of the following shows the correct relationship between the pressure $^{'} P^{'}$ and density $ρ$ of an ideal gas at constant temperature ?