Question

Assuming the expression for the pressure exerted by the gas on the walls of the container, it can be shown that pressure is

• A. $\left[\frac{1}{3}\right]^{rd}$ kinetic energy per unit volume of a gas
• B. $\left[\frac{2}{3}\right]^{rd}$ kinetic energy per unit volume of a gas
• C. $\left[\frac{3}{4}\right]^{th}$ kinetic energy per unit volume of a gas
• D. $\frac{3}{3} \times$ kinetic energy per unit volume of a gas

Hint:

Recall the formula of the rms speed of a molecule of gas and the ideal gas equation.

Answer 1

The Correct Option is B

The kinetic energy of the gas is given by

\(K =\frac{1}{2} mv _{ rms }^{2}\)

Where

• \(v_{rms}\) is the root mean square (rms) velocity of the molecules of the gas
• \(m\) is the mass of one molecule of the gas

\(R m s\) velocity of gas molecules is given by

 \(v _{ rms }=\sqrt{\frac{3 R T}{m}}\) 

Where

• T is the absolute temperature
• R is the Universal gas constant

\(\therefore K =\frac{1}{2} m \cdot \frac{3 RT }{ m }=\frac{3}{2} RT\) 
 

From the ideal gas equation, \(RT = PV\)
\(\Rightarrow K =\frac{3}{2} PV\) 
 

Thus we get the pressure of the gas \(P =\frac{2}{3} \frac{ K }{ V }\)

Therefore, the pressure exerted by the gas on the walls of the container is \([\frac{2}{3}]^{rd}\) kinetic energy per unit volume of a gas.

Discover More from Chapter: Kinetic Theory of Gases

Answer 2

The Correct Answer is (B)

Real Life Applications

1. Due to the collision of the air molecules with the walls of the tire results in the pressure of the air in a tire. 
2. The pressure of water in a dam is caused by the weight of the water. 
3. The pressure of blood in arteries is caused by the heart. 
4. The pressure of the atmosphere is caused by the weight of the air.

Question can also be asked as

1. What is the relationship between pressure and the number of gas molecules? 
2. How does the pressure of a gas change with the volume of the container? 
3. What is the effect of temperature on the pressure of a gas?
 

Answer 3

The Correct Answer is (B)

According to the Kinetic theory of gases

• A given amount of gas is a mixture of a large number of identical molecules.
• The molecules of the gas move randomly in all directions.
• At ordinary temperature and pressure, the size of the molecules is very small compared to the distance between them.
• The molecules do not exert any force of attraction or repulsion on each other.
• The collision of the molecules against each other or with the walls of the container is perfectly elastic.

Pressure of an Ideal Gas

The pressure exerted by an ideal gas is given by the formula

\(P=\frac{1}{3}\frac{M}{V}v^2\)

Where

• M is the mass of the gas
• V is the volume of the gas
• v is the rms velocity of the molecules of the gas

RMS Speed

• The RMS speed or root mean square speed is defined as the square root of the mean of the squares of the random speeds of the individual molecules of the gas.
• The formula for the RMS speed of the molecules of the gas is \(v_{rms}=\sqrt{\frac{3k_B T}{m}}\)

Related Topics
Pressure of an Ideal gas	Kinetic Theory	Kinetic Theory Important Questions
Kinetic Interpretation of Temperature	Kinetic Theory of Gas Formula	Kinetic Theory MCQs