Question

The expression for total kinetic energy per unit volume of gas is,

A.)$\dfrac{E}{V}=\dfrac{p}{2}$
B.)$\dfrac{E}{V}=\dfrac{p}{3}$
C.)$\dfrac{E}{V}=\dfrac{2p}{3}$
D.)$\dfrac{E}{V}=\dfrac{3p}{2}$

Answer 1

Hint: Study about the kinetic theory of gases. Try to obtain the mathematical expression for the kinetic energy of gas and study the ideal gas law. With the help of ideal gas law we can find the solution for this question.

Complete step by step answer:
The kinetic molecular theory of gases is a model which helps us to understand the physical properties of gases at the molecular level.

Gases consist of small particles which are always constant random motions. There are no interactive forces acting between the particles in a gas. The average kinetic energy of gas particles is directly proportional to the absolute temperature the gas is in. The kinetic energy of all gases at the same constant temperature is the same.

Kinetic energy of individual atoms or molecules can be defined as, $KE=\dfrac{1}{2}m{{v}^{2}}$

According to the kinetic molecular theory of gases, the average kinetic energy of gases can be expressed as,

${{E}_{k}}=\dfrac{3}{2}kT$

It can also be expressed as,

$E=\dfrac{3}{2}nRT$

Again, for an ideal gas we have ideal gas law which gives that,

$PV=nRT$

Putting this value on the above equation, we get that,

$\begin{aligned} & E=\dfrac{3}{2}PV \\\ & \dfrac{E}{V}=\dfrac{3}{2}P \\\ \end{aligned}$

So, we get that the total kinetic energy per unit volume of a gas can be expressed as,
$\dfrac{E}{V}=\dfrac{3}{2}P$

The correct option is (D).

Note: here we don’t have to calculate anything. just obtain the equation and substitute the value of nRT from ideal gas law to get the energy per unit volume of gas in terms of pressure. Try to look for this type of alternative theories which can be used to change the terms in a given equation.