Question

The relation between the internal energy U and adiabatic constant $\gamma$ is
A. $U=\dfrac{PV}{\gamma -1}$
B.$U=\dfrac{P{{V}^{\gamma }}}{\gamma -1}$
C. $U=\dfrac{PV}{\gamma }$
D. $U=\dfrac{\gamma }{PV}$

Answer 1

As a first step, you could recall the expression for change in internal energy and assume that initially the temperature and internal energy of the system is zero. Now you could recall the expression for specific heat at constant volume in terms of heat capacity ratio$\gamma$ and also the ideal gas equation. Then, you could accordingly substitute to get the required expression.

Formula used: Expression for change in internal energy,
$\Delta U={{C}_{V}}\Delta T$
Expression for heat capacity at constant volume,
${{C}_{V}}=\dfrac{nR}{\gamma -1}$
Ideal gas equation,
$PV=nRT$

Complete step by step answer:
In the question, we are asked to find the relation between the internal energy U and adiabatic constant $\gamma$ among the given options.
We could derive the required relation from the expression for change in internal energy given by,
$\Delta U={{C}_{V}}\Delta T$
$\Rightarrow {{U}_{2}}-{{U}_{1}}={{C}_{V}}\left( {{T}_{2}}-{{T}_{2}} \right)$ ………………………………. (1)
Where, ${{U}_{1}}$ and${{U}_{2}}$ are internal energies of initial and final conditions, ${{C}_{V}}$ is the specific heat at constant volume and ${{T}_{1}}$ and${{T}_{2}}$ are the initial and final temperatures.
Let us take the initial state of the system under consideration to have zero temperature and zero internal energy. That is, let us assume the initial conditions of the system were,
${{U}_{1}}=0$
${{T}_{1}}=0$
Then,
${{U}_{2}}=U$
${{T}_{2}}=T$
Then, we could say that (1) will be,
$U={{C}_{V}}T$ ………………………………… (2)
Now let us recall the ideal gas equation which is given by,
$PV=nRT$
$\Rightarrow T=\dfrac{PV}{nR}$ …………………………. (3)
Where, P is the pressure, V is the volume and R is the ideal gas constant.
We also have the relation between the heat capacity at constant volume and the heat capacity ratio$\gamma$ which is given by,
${{C}_{V}}=\dfrac{nR}{\gamma -1}$ ……………………………………….. (4)
Substituting (3) and (4) in (2), we get,
$U=\left( \dfrac{nR}{\gamma -1} \right)\left( \dfrac{PV}{nR} \right)$
$\therefore U=\dfrac{PV}{\gamma -1}$
Therefore, we found the relation between the internal energy U and adiabatic constant $\gamma$ to be,
$U=\dfrac{PV}{\gamma -1}$

So, the correct answer is “Option A”.

Note: The adiabatic constant $\gamma$ is called by many names. By definition, it is actually the ratio of specific heats or, the heat capacity ratio. It is also called adiabatic index or Laplace coefficient. It is also called the isentropic expansion factor. This constant has numerous applications in thermodynamic processes involving ideal gases.