A monatomic ideal gas sample is given heat Q . One half of this heat is used as work done by the gas and rest is used for increasing its internal energy. The equation of process in terms of volume and temperature is

Question

A monatomic ideal gas sample is given heat Q . One half of this heat is used as work done by the gas and rest is used for increasing its internal energy. The equation of process in terms of volume and temperature is (A) (V2/T3)= constant (B) (V2/√T)= constant (C) VT3= constant (D) V2√T= constant

Tardigrade Admin · Accepted Answer

PdV=nCvdT ⇒ (nRT/V)dV=nCvdT ⇒ (dV/V)=(3/2)(dT/T) ∫ (dV/V)=(3/2)∫ (dT/T) ∴ V2=CT3 , Where C is a constant.

Q. A monatomic ideal gas sample is given heat $Q$ . One half of this heat is used as work done by the gas and rest is used for increasing its internal energy. The equation of process in terms of volume and temperature is

$\frac{V ^{2}}{T ^{3}} =$ constant

$\frac{V ^{2}}{T} =$ constant

$V T^{3} =$ constant

$V^{2} T =$ constant

$P d V = n C_{v} d T$
$\Rightarrow \frac{n RT}{V} d V = n C_{v} d T$
$\Rightarrow \frac{d V}{V} = \frac{3}{2} \frac{d T}{T}$
$\int \frac{d V}{V} = \frac{3}{2} \int \frac{d T}{T}$
$∴ V^{2} = C T^{3}$ , Where $C$ is a constant.

Q. A monatomic ideal gas sample is given heat Q . One half of this heat is used as work done by the gas and rest is used for increasing its internal energy. The equation of process in terms of volume and temperature is

T3V2​= constant

T​V2​= constant

VT3= constant

V2T​= constant