Question

During an adiabatic process the square of the pressure of a gas is proportional to the fifth power of its absolute temperature. The ratio of specific heat cp / cv for that gas is

Answer 1

Answer:

Given:

For an Adiabatic process, the square of the pressure is proportional to the 5th power of Temperature.

To find:

Ratio of Cp and Cv.

Calculation:

As per the question:

${P}^{2} \propto {T}^{5}$

$= > {P} \propto {T}^{( \frac{5}{2}) }$

Now, considering Ideal Gas Equation :

$= > {P} \propto { (\frac{PV}{nR}) }^{( \frac{5}{2}) }$

$= > {P} \propto {( PV )}^{( \frac{5}{2}) }$

$= > {P}^{ \frac{3}{2} } {V}^{ \frac{5}{2} } = constant$

$= > P {V}^{ \frac{5}{3} } = constant$

Now comparing with standard Adiabatic process :

$P {V}^{ \gamma } = constant$

So, from comparison , we get

$\gamma = \frac{5}{3}$

We know that value of γ is equal to Cp/Cv

Hence :

$\frac{Cp}{Cv} = \frac{5}{3}$

Answer 2

Answer: 5/3

Explanation:

Let the pressure of the gas be P and temperature be T.  Cp/Cv is a constant ration and denoted by λ. i.e.

$\frac{C_p}{C_v}=\lambda$

We know that relation between P, T and λ is given by

$T.P^\frac{1-\lambda}{\lambda}=\text{Constant}\;\;\;\;\;\;\;...........i)$

According  to question,

Square of pressure ∝ Fifth power of Temperature

$P^2\propto T^5\\\;\\P^\frac{2}{5}\propto T\\\;\\\frac{P^\frac{2}{5}}{T}=\text{Constant}\\\;\\\frac{T}{P^\frac{2}{5}}=\text{Constant}\\\;\\T.P^{(-\frac{2}{5})}=\text{Constant}$

Comparing above equation with equation i),

We have,

$\frac{1-\lambda}{\lambda}=-\frac{2}{5}\\\;\\5(1-\lambda)=-2\lambda\\\;\\5-5\lambda=-2\lambda\\\;\\5=5\lambda-2\lambda\\\;\\5=3\lambda\\\;\\\lambda=\frac{5}{3}\\\;\\\frac{C_p}{C_v}=\frac{5}{3}$