Question

11. The slopes of isothermal and adiabatic curves are related as:
(A) Isothermal curve slope = Adiabatic curve slope.
(B) Isothermal curves slope = Yx Adiabatic curve slope.
(C) Adiabatic curve slope = Yx isothermal curve slope.
(D) Adiabatic curve slope = 12x isothermal curve slope.​

Answer 1

Hi there!

Explanation:

Slope of Isothermal Graph is (- p/v)

Slope of Adiabatic Graph is (- y p/v)

→ (- y p/v) = y × (- p/v)

→ Adiabatic Graph = y × Isothermal Graph

Answer:

Option (C)

Answer 2

$\orange\bigstar\:{\underline{\pink{\boxed{\bf{\gray{(C)\:Adiabatic\:curve\:slope\:=\:\atop{\gamma\times{Isothermal\:curve\:slope}}}}}}}}\:\green\bigstar$

$\Large{\purple{\underline{\textsf{\textbf{Step-by-step\:Explanation\::}}}}}$

❶ Isᴏ-ᴛʜᴇʀᴍᴀʟ Pʀᴏᴄᴇss :-

✯ It is a process that takes place in the system under constant constant temperature.

★ The equation that represents this process is,

➛ $\bf{P\:V\:=\:C}$

Wʜᴇʀᴇ,

• P is the pressure.

• V is the volume.

• C is a constant term.

↝ Differentiating w.r.t V,

➛ $\sf{V\:\dfrac{dP}{dV}\:+\:P\:\dfrac{dV}{dV}\:=\:0}$

➛ $\sf{V\:\dfrac{dP}{dV}\:=\:-\:P\:\dfrac{dV}{dV}\:}$

➛ $\bf\blue{\dfrac{dP}{dV}\:=\:-\:\dfrac{P}{V}\:}$

$\Large\bf{Therefore,}$

The slope of isothermal process is $\tt{-\dfrac{P}{V}}$.

❷ Aᴅɪᴀʙᴀᴛɪᴄ Pʀᴏᴄᴇss :-

✯ It is a process which takes place with zero heat transfer from the system or surroundings.

★ The equation that represents this process is,

➙ $\bf{P\:V^{\gamma}\:=\:C}$

Wʜᴇʀᴇ,

• γ is the ratio of specific heat calculated at constant pressure and constant volume.

↝ Differentiating w.r.t V,

➙ $\sf{V^{\gamma}\:\dfrac{dP}{dV}\:+\:P\:\dfrac{d(V^{\gamma})}{dV}\:=\:0}$

➙ $\sf{V^{\gamma}\:\dfrac{dP}{dV}\:=\:-\:P\:\dfrac{d(V^{\gamma})}{dV}\:}$

➙ $\sf{V^{\gamma}\:\dfrac{dP}{dV}\:=\:-\:P\:{\gamma}\:V^{(\gamma\:-\:1)}\:}$

➙ $\sf{\dfrac{dP}{dV}\:=\:\dfrac{-\:P\:{\gamma}\:V^{(\gamma\:-\:1)}}{V^{\gamma}}\:}$

➙ $\sf{\dfrac{dP}{dV}\:=\:-\:P\:{\gamma}\:V^{(\gamma\:-\:1\:-\:\gamma)}\:}$

➙ $\sf{\dfrac{dP}{dV}\:=\:-\:P\:{\gamma}\:V^{-1}\:}$

➙ $\bf\pink{\dfrac{dP}{dV}\:=\:-\:{\gamma}\:\dfrac{P}{V}\:}$

$\Large\bf{Therefore,}$

The slope of adiabatic process is $\tt{-\gamma\dfrac{P}{V}}$.

Nᴏᴡ,

➣ Comparing the slope, we see that the slope of the adiabatic curve is γ times the slope of the isothermal curve.