Question

Two ideal gases have the same value of Cp / Cv = γ. What will be the value of this ratio for a mixture of the two gases in the ratio 1 2?

Answer 1

Explanation:

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Q. 11

Two ideal gases have the same value of CP/CV = γ. What will be the value of this ratio for a mixture of the two gases in the ratio 1: 2?

Class 12thHC Verma - Concepts of Physics Part 227. Specific Heat Capacities of Gases

Answer

We know Cp/Cv=γ, R=Cp-Cv,

where the molar heat capacity C, at constant pressure, is represented by Cp, at constant volume, the molar heat capacity C is represented by Cv and R is the universal gas constant.

Now,



For the first ideal gas,





Where Cp1 and CV1 is the molar heat capacity at constant pressure and constant volume

Similarly, for the second ideal gas





Where Cp2 and CV2 is the molar heat capacity at constant pressure and constant volume

Given,



i.e

dU1=nCV1dT

dU2=2nCV2Dt

When gas is mixed,







Also,



From (1) and (2)



Answer 2

Explanation:

Step 1:

In case of  first ideal gas,

$C_{p 1}=$ constant pressure  specific heat  

$C_{V 1}=$ constant volume specific heat  

$n_{1}=$ Amount of Gas Moles

$\frac{c_{p 1}}{C_{v 1}}=\mathrm{y} \text { and } \mathrm{C}_{\mathrm{p} 1}-\mathrm{C}_{\mathrm{v} 1}=R$

γ $C_{\mathrm{v} 1}-C_{\mathrm{v} 1}=R$

$C_{v 1}$ (γ − 1 )$=R$

$C_{v 1}=\frac{R}{\gamma-1}$

$C_{p 1}=\gamma \frac{R}{\gamma-1}$

Step 2:

In case of  second ideal gas,

$C_{p 2}=$  constant pressure specific heat  

$C_{v 2}=$ constant volume specific heat  

$n_{2}=$ Amount of Gas Moles

$\frac{C_{p 2}}{C_{v 2}}$= γ and $C_{\mathrm{p} 2}-C_{\mathrm{v} 2}=R$

γ$C_{v 2}-C_{v 2}=R$

$C_{v 2}$ (γ − 1) $=R$

$C_{v 2}=\frac{R}{\gamma-1}$

$C_{p 2}=\gamma \frac{R}{\gamma-1}$

Step 3:

Given in the question,

$n_{1}=n_{2}=1: 2$

$d U_{1}=n C_{\mathrm{v} 1} d t$

$d U_{2}=2 n C_{v 2} d T$

Step 4:

When mixing the gasses,

$n C_{v 1} d T+2 n C_{v 2} d T=3 n C_{v} d T$

$C_{v}=\frac{C_{v 1}+2 C_{v 2}}{3}$

$C_{v}=\frac{\frac{R}{\gamma-1}+2 \frac{R}{\gamma-1}}{3}$

$C_{v}=\frac{3 R}{(\gamma-1)^{3}}=\frac{R}{\gamma-1}$

therefore , in the mixture $\frac{C_{v}}{C_{p}}$ is γ.