(C) For an isothermal process,$PV = \text{constant}$. Differentiating with respect to $V$,we get $P + V(dP/dV) = 0$,so the slope $(dP/dV)_{\text{iso}}

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Slope of adiabatic curve $= \gamma \times$ Slope of isothermal curve

(C) For an isothermal process,$PV = \text{constant}$. Differentiating with respect to $V$,we get $P + V(dP/dV) = 0$,so the slope $(dP/dV)_{\text{iso}} = -P/V$.For an adiabatic process,$PV^{\gamma} = \text{constant}$. Differentiating with respect to $V$,we get $P(\gamma V^{\gamma-1}) + V^{\gamma}(dP/dV) = 0$.Rearranging gives $(dP/dV)_{\text{adia}} = -\gamma P/V$.Comparing the two,we find that $(dP/dV)_{\text{adia}} = \gamma \times (dP/dV)_{\text{iso}}$.

Class 11 Physics Thermodynamics Mix Examples-Thermodynamics

Medium

What is the relationship between the slopes of isothermal and adiabatic curves?

A
Slope of isothermal curve $=$ Slope of adiabatic curve
B
Slope of isothermal curve $= \gamma \times$ Slope of adiabatic curve
C
Slope of adiabatic curve $= \gamma \times$ Slope of isothermal curve
D
Slope of adiabatic curve $= 7 \times$ Slope of isothermal curve

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Mix Examples-Thermodynamics

Match List-$I$ with List-$II$:

List-$I$ List-$II$

$(a)$ Isothermal $(i)$ Pressure constant

$(b)$ Isochoric $(ii)$ Temperature constant

$(c)$ Adiabatic $(iii)$ Volume constant

$(d)$ Isobaric $(iv)$ Heat content is constant

Choose the correct answer from the options given below:

List-$I$	List-$II$
$(a)$ Isothermal	$(i)$ Pressure constant
$(b)$ Isochoric	$(ii)$ Temperature constant
$(c)$ Adiabatic	$(iii)$ Volume constant
$(d)$ Isobaric	$(iv)$ Heat content is constant

MediumJEE MAIN 2021

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Two gases have the same initial pressure,volume,and temperature. They expand to the same final volume,one adiabatically and the other isothermally. Which of the following statements is correct regarding the final state?

Difficult

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An ideal gas is made to undergo a cycle depicted by the $p-V$ diagram given below. The curved line from $A$ to $B$ is an adiabat. Then,

MediumKVPY 2010

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The efficiency of the cycle shown in the figure (consisting of one isobar,one adiabat,and one isotherm) is $50 \%$. The ratio $x$ between the highest and lowest temperatures attained in this cycle obeys (the working substance is an ideal gas):

KVPY 2020

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$A$ bubble containing $8$ moles of Helium is submerged at a certain depth in water. When the temperature of the water increases by $30 \ ^\circ C$,how much heat in $J$ is added during the expansion of the Helium bubble?

Medium

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What is the relationship between the slopes of isothermal and adiabatic curves?

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Match List-$I$ with List-$II$: List-$I$ List-$II$ $(a)$ Isothermal $(i)$ Pressure constant $(b)$ Isochoric $(ii)$ Temperature constant $(c)$ Adiabatic $(iii)$ Volume constant $(d)$ Isobaric $(iv)$ Heat content is constant Choose the correct answer from the options given below:

Two gases have the same initial pressure,volume,and temperature. They expand to the same final volume,one adiabatically and the other isothermally. Which of the following statements is correct regarding the final state?

An ideal gas is made to undergo a cycle depicted by the $p-V$ diagram given below. The curved line from $A$ to $B$ is an adiabat. Then,

The efficiency of the cycle shown in the figure (consisting of one isobar,one adiabat,and one isotherm) is $50 \%$. The ratio $x$ between the highest and lowest temperatures attained in this cycle obeys (the working substance is an ideal gas):

$A$ bubble containing $8$ moles of Helium is submerged at a certain depth in water. When the temperature of the water increases by $30 \ ^\circ C$,how much heat in $J$ is added during the expansion of the Helium bubble?

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Match List-$I$ with List-$II$:

List-$I$ List-$II$

$(a)$ Isothermal $(i)$ Pressure constant

$(b)$ Isochoric $(ii)$ Temperature constant

$(c)$ Adiabatic $(iii)$ Volume constant

$(d)$ Isobaric $(iv)$ Heat content is constant

Choose the correct answer from the options given below: