Assertion The total translational kinetic energy of all the molecules of a given mass of an ideal ga

Q: Why is the translational kinetic energy $\frac{3}{2} nRT$?

According to the equipartition theorem, each degree of freedom contributes $\frac{1}{2} kT$ per molecule. A monatomic gas has 3 translational degrees of freedom, giving $\frac{3}{2} kT$ per molecule, or $\frac{3}{2} nRT$ for $n$ moles.

Q: Does the reason statement imply the assertion is false?

No, the reason statement is factually correct. Molecules do collide and change velocities. However, it is just a postulate and doesn't serve as the mathematical proof for the $1.5 pV$ relation.

Q: What if the gas was diatomic? Would the assertion still hold?

The assertion specifically mentions *translational* kinetic energy. Regardless of the gas being monatomic, diatomic, or polyatomic, the *translational* kinetic energy is always $\frac{3}{2} nRT = 1.5 pV$. The *total* kinetic energy would be different for diatomic gases due to rotational degrees of freedom.

Q: Why do we use $pV = nRT$ here?

The assertion relates kinetic energy to the product of pressure and volume ($pV$). Since $E$ is naturally expressed in terms of $T$ ($E = \frac{3}{2} nRT$), the ideal gas law is the perfect bridge to convert $nRT$ into $pV$.

Q: What would be the correct reason for this assertion?

A correct reason would be: "According to the kinetic theory of gases, the pressure exerted by an ideal gas is $p = \frac{1}{3} \frac{mN}{V} v_{rms}^2$, and the total translational kinetic energy is $E = \frac{1}{2} mN v_{rms}^2$, which mathematically leads to $E = \frac{3}{2} pV$."

Question

**Assertion** The total translational kinetic energy of all the molecules of a given mass of an ideal gas is 1.5 times the product of its pressure and its volume.
**Reason** The molecules of a gas collide with each other and the velocities of the molecules change due to the collision.

Options:

(A) If Assertion is true, Reason is true; Reason is the correct explanation for Assertion

(B) If Assertion is true, Reason is true; Reason is not a correct explanation for Assertion

(C) If Assertion is true; Reason is false

(D) If Assertion is false; Reason is true

Sigma Percentile · Accepted Answer

The problem we are tackling today is a classic Assertion-Reason question from the Kinetic Theory of Gases. These questions are notorious for testing not just your memory of formulas, but your deep conceptual clarity and logical reasoning. Let's break it down step-by-step and uncover the physics hidden within!

### Decoding the Assertion

The assertion states: **"The total translational kinetic energy of all the molecules of a given mass of an ideal gas is 1.5 times the product of its pressure and its volume."**

To verify this, we need to recall the fundamental expression for the translational kinetic energy of an ideal gas. From the kinetic theory, the total translational kinetic energy $E$ for $n$ moles of an ideal gas is given by:

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

This beautiful equation tells us that the kinetic energy depends solely on the absolute temperature $T$ of the gas. But the assertion talks about pressure $p$ and volume $V$. How do we bridge this gap?

Enter the Ideal Gas Law! We know that for an ideal gas:

$$pV = nRT$$

By substituting $nRT$ with $pV$ in our energy equation, we get:

$$E = \frac{3}{2} pV$$

Since $\frac{3}{2}$ is exactly $1.5$, we can rewrite this as:

$$E = 1.5 pV$$

This perfectly matches the statement given in the assertion. Therefore, we can confidently say that the **Assertion is absolutely True**.

### The Kinetic Theory Connection

Now, let's shift our focus to the reason provided: **"The molecules of a gas collide with each other and the velocities of the molecules change due to the collision."**

Is this a factual statement? Absolutely. According to the postulates of the kinetic theory of gases, gas molecules are in a state of continuous, random motion. They constantly collide with each other and with the rigid walls of their container. These collisions are assumed to be perfectly elastic.

Because of these incessant collisions, the individual velocity of any given molecule is constantly changing in both magnitude and direction. So, the **Reason is also a True statement**.

### Evaluating the Logical Link

Here is where the real test lies. Both statements are true, but does the reason actually *explain* the assertion?

The relation $E = 1.5 pV$ arises mathematically from the fact that a gas has three degrees of translational freedom (moving in x, y, and z directions), each contributing $\frac{1}{2} kT$ to the energy. The derivation of pressure $p = \frac{1}{3} \frac{mN}{V} v_{rms}^2$ and energy $E = \frac{1}{2} mN v_{rms}^2$ leads to this exact factor of $1.5$.

While collisions are a fundamental part of the kinetic theory and are responsible for maintaining thermal equilibrium, the mere fact that "velocities change due to collisions" does not mathematically or logically explain *why* the kinetic energy is exactly $1.5$ times the product of pressure and volume.

### The Final Verdict

Since both the assertion and the reason are true, but the reason does not provide the correct explanation for the assertion, the correct choice is **Option (b)**.

Always remember, in Assertion-Reason questions, finding both statements to be true is only half the battle. The critical step is to ask yourself: "Does statement B directly cause or mathematically prove statement A?" If the answer is no, you know exactly what to choose!

Select Answer:

Visualized Solution

$A na l y z in g t h e A sser t i o n$

$F or m u l a f or K in e t i c E n er g y$

$I d e a l G a s E q u a t i o n$

$V er i f y in g t h e A sser t i o n$

$A na l y z in g t h e R e a so n$

$K in e t i c T h eor y P os t u l a t es$

$L ink in g A sser t i o n an d R e a so n$

$F ina l C o n c l u s i o n$

The Sigma Insight: Kinetic Theory of Gases

Decoding the Assertion

The Kinetic Theory Connection

Evaluating the Logical Link

The Final Verdict

Similar Questions

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On the basis of kinetic theory of gases, the gas exerts pressure because its molecules

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Visualized Solution

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Analyzing the Assertion

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Formula for Kinetic Energy

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Ideal Gas Equation

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Verifying the Assertion

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Analyzing the Reason

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Kinetic Theory Postulates

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Linking Assertion and Reason

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Final Conclusion