Question

The change in internal energy of an ideal gas when volume changes from V to 2V at constant pressure P is:
a.) $\dfrac{R}{y-1}$
b.) $PV$
c.) $\dfrac{PV}{y-1}$
d.) $\dfrac{yPV}{y-1}$

Answer 1

Hint: The ideal gases are hypothetical gases which follows a few rules which includes that the ideal gases do not attract or repel each other. The only interaction which is between the ideal gas molecule would be an elastic collision upon impact with each other or in an elastic Collision between the walls of the container.

Complete Solution :
Let the number of moles of gas in the container be n, by using the ideal gas equation, we can say that:
PV=nRT--(1)
Where p= pressure of the system, V = volume of the system, T = temperature on an absolute scale and n= number of moles of gas in the container
And it is given that the volume changes from V to 2V and the pressure remains constant.
The change in internal energy will be =$U=n\times {{C}_{V}}\times T$ --(2)

By using the equation (1) and (2) we can say that:
\[\begin{align}
  & U={{C}_{V}}\times \dfrac{P\times V}{R}=\dfrac{{{C}_{v}}\times P\times V}{{{C}_{p}}-{{C}_{V}}} \\
 & =\dfrac{P\times V}{\dfrac{{{C}_{p}}}{{{C}_{V}}}-1}=\dfrac{PV}{y-1} \\
\end{align}\]
Hence, the correct answer is the change in internal energy of an ideal gas when volume changes from V to 2V at constant pressure P is $\dfrac{PV}{y-1}$.
So, the correct answer is “Option C”.

Note: The specific heat is defined as the amount of heat per unit mass which is required to raise the temperature of a system by one degree centigrade. This relationship does not apply if there is a phase change between systems because the heat which is added or removed during the process of phase change of a system does not change the temperature of the system.