Question

During an adiabatic process, the pressure of a gas is found to be proportional to the cube of its absolute temperature. The ratio $C_p/C_{\nu }$ for the gas is

• A.
  
• B.
  
  
• C.
• D.
  
  

Answer 1

Step 1: Given that:

In the adiabatic process,

$P\propto T^3$

Step 2: Formula used

$P{V}^{\text{γ}}=constant\left(Adiabaticprocess\right)$

$PV=nRT$

Step 3: Calculation of $\frac{C_P}{C_v}$ :

In adiabatic process, the gas equation is given by;

$P{V}^{\text{γ}}=constant$

Where, γ is the ratio of specific heat at constant pressure and constant volume, that is;

γ= $\frac{C_P}{C_V}$

Also,

$PV=nRT$

$V=\frac{nRT}{P}$

Therefore,

$P{\left(\frac{nRT}{P}\right)}^{\text{γ}}=constant$

$P^{1-\text{γ}}{T}^{\text{γ}}=constant$

$P{T}^{\frac{\text{γ}}{1-\text{γ}}}=constant$

$P\propto T^{\frac{\text{γ}}{1-\text{γ}}}$

Also, it is given that

$P\propto T^3$

Thus,

Comparing the equation of gas in adiabatic process with the given relation, we get;

$\frac{\text{γ}}{\text{γ}-1}=3$

$3\text{γ}-3=\text{γ}$

$3\text{γ}-\text{γ}=3$

$2\text{γ}=3$

$\text{γ}=\frac{3}{2}$

Thus,

The value of $\frac{C_P}{C_V}$ will be $\frac{3}{2}$ .

Therefore,

The correct option is B) .