Question

If the ratio of sp.heat of a gas at constant pressure to that at constant volume is , the change in internal energy of gas, when the volume changes from V to 2V at constant pressure P is
A. $\dfrac{R}{\gamma -1}$
B. PV
C. $\dfrac{PV}{\gamma -1}$
D. $\dfrac{\gamma PV}{\gamma -1}$

Answer 1

Hint: Charles’s law states that the volume of an ideal gas at constant pressure is directly proportional to the absolute temperature. The ratio of the heat capacity at constant pressure (CP) to heat capacity at constant volume (CV) is known as adiabatic index. It is sometimes also known as the isentropic expansion factor and is denoted by γ (gamma).Ideal gas refers to a hypothetical gas composed of molecules which follow a few rules: Ideal gas molecules do not attract or repel each other. The only interaction between ideal gas molecules would be an elastic collision upon impact with each other or an elastic collision with the walls of the container.

Solution Let the number of moles of gas taken is n
So, by ideal gas equation,
P$\times $V=n$\times $R$\times $T
As the volume changes from V to 2V.
Where pressure P remains Constant. We can say ?V=V and P$\times $?V=n$\times $R$\times $?T
P$\times $V=n$\times $R$\times $?T..... (1), where ?T represents the rise in temperature on an absolute scale.
Again, change in internal energy
?U=n$\times $Cv$\times $?T.......(2)
By (1) and (2)
?U=Cv$\times $$\dfrac{P\times V}{R}$ = $\dfrac{(Cv\times P\times
V)}{(Cp-Cv)}$
= $\dfrac{P\times V}{\dfrac{Cp}{Cv}-1}$ = $\dfrac{PV}{\gamma -1}$ where
$\dfrac{Cp}{Cv}=$ $\gamma $

Hence, Correct option is C.

Note: The specific heat is the amount of heat per unit mass required to raise the temperature by one degree Celsius.. The relationship does not apply if a phase change is encountered, because the heat added or removed during a phase change does not change the temperature.