For a diatomic gas change in internal energy for unit change in temperature for constant pressure and constant volume is Δ U1 and Δ U2 respectively. The ratio of Δ U1: Δ U2 is

Question

For a diatomic gas change in internal energy for unit change in temperature for constant pressure and constant volume is Δ U1 and Δ U2 respectively. The ratio of Δ U1: Δ U2 is (A) 5: 3 (B) 3: 5 (C) 1: 1 (D) 5: 7

Tardigrade Admin · Accepted Answer

According to first law of thermodynamics,

Δ Q = Δ U + Δ W

At constant pressure,

Δ Q = n C_{P} Δ T, Δ U = Δ U_{1}

,

Δ W = P Δ V = n R Δ T

∴ n C_{P} Δ T = Δ U + n R Δ T

or

n (C_{P} - R) Δ T = Δ U_{1}

At constant volume,

Δ Q = n C_{V} Δ T, Δ W = 0

∴ n C_{V} Δ T = Δ U_{2}

Thus,

\frac{Δ U _{1}}{Δ U _{2}} = \frac{C _{P} - R}{C _{V}} = \frac{C _{V}}{C _{V}} = 1 : 1

Q. For a diatomic gas change in internal energy for unit change in temperature for constant pressure and constant volume is $Δ U_{1}$ and $Δ U_{2}$ respectively. The ratio of $Δ U_{1} : Δ U_{2}$ is

5 : 3

3 : 5

1 : 1

5 : 7

Q. For a diatomic gas change in internal energy for unit change in temperature for constant pressure and constant volume is ΔU1​ and ΔU2​ respectively. The ratio of ΔU1​:ΔU2​ is

5 : 3

3 : 5

1 : 1

5 : 7

According to first law of thermodynamics, ΔQ=ΔU+ΔW At constant pressure, ΔQ=nCP​ΔT,ΔU=ΔU1​, ΔW=PΔV=nRΔT ∴nCP​ΔT=ΔU+nRΔT or n(CP​−R)ΔT=ΔU1​ At constant volume, ΔQ=nCV​ΔT,ΔW=0 ∴nCV​ΔT=ΔU2​ Thus, ΔU2​ΔU1​​=CV​CP​−R​=CV​CV​​=1:1

Q. For a diatomic gas change in internal energy for unit change in temperature for constant pressure and constant volume is $Δ U_{1}$ and $Δ U_{2}$ respectively. The ratio of $Δ U_{1} : Δ U_{2}$ is