An ideal gas is cooled from 400 K to 324 K. What is the decrease in urms of the gas?

The correct answer is Option 4. Full mathematical solution steps and hints are accessible in the interactive StemCET workspace.

Chemistry Q126 – AP-EAMCET 2026

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Home Chemistry Kinetic TheoryQuestion #449

An ideal gas is cooled from

400 K

324 K

. What is the decrease in

u_{r m s}

of the gas?

Clue 1 of 3: Root Mean Square Velocity and Temperature

The root mean square (RMS) velocity of gas molecules is directly related to the absolute temperature of the gas. Understanding this relationship is key to solving the problem.

🧠 Full Solution Path

Step 1: Establish the relationship between

u_{r m s}

and temperature✦ Active

The root mean square velocity ( $u_{r m s}$ ) of an ideal gas is directly proportional to the square root of its absolute temperature ( $T$ ). This relationship is given by the formula:

u_{r m s} \propto \sqrt{T}

Therefore, the ratio of $u_{r m s}$ at two different temperatures $T_{1}$ and $T_{2}$ can be written as:

\frac{u_{r m s, 2}}{u_{r m s, 1}} = \sqrt{\frac{T_{2}}{T_{1}}}

💡 Teacher's Secret Hint

Remember that temperature must always be in Kelvin (absolute temperature) when using gas laws and kinetic theory formulas. The problem provides temperatures in Kelvin, so no conversion is needed.

Step 2: Calculate the ratio of final to initial

u_{r m s}

○ Expand

Given initial temperature $T_{1} = 400 K$ and final temperature $T_{2} = 324 K$ . Substitute these values into the ratio formula:

\frac{u_{r m s, 2}}{u_{r m s, 1}} = \sqrt{\frac{324 K}{400 K}}

Simplify the square root:

\frac{u_{r m s, 2}}{u_{r m s, 1}} = \sqrt{\frac{18^{2}}{20^{2}}} = \frac{18}{20} = \frac{9}{10} = 0.9

This means the final $u_{r m s}$ is $0.9$ times the initial $u_{r m s}$ , or $u_{r m s, 2} = 0.9 \cdot u_{r m s, 1}$ .

💡 Teacher's Secret Hint

Simplifying fractions under the square root before taking the root can sometimes make calculations easier. Recognize perfect squares like $324 = 18^{2}$ and $400 = 20^{2}$ .

Step 3: Calculate the percentage decrease in

u_{r m s}

○ Expand

The decrease in $u_{r m s}$ is the initial value minus the final value:

Decrease = u_{r m s, 1} - u_{r m s, 2}

Substitute $u_{r m s, 2} = 0.9 \cdot u_{r m s, 1}$ :

Decrease = u_{r m s, 1} - 0.9 \cdot u_{r m s, 1} = 0.1 \cdot u_{r m s, 1}

To express this as a percentage decrease, divide by the initial value and multiply by $100 %$ :

Percentage Decrease = \frac{Decrease}{u_{r m s, 1}} \times 100 %

Percentage Decrease = \frac{0.1 \cdot u_{r m s, 1}}{u_{r m s, 1}} \times 100 % = 0.1 \times 100 % = 10 %

💡 Teacher's Secret Hint

Always ensure you divide by the initial value when calculating percentage decrease or increase. A common mistake is to divide by the final value.

✦ STEM Console utilizes AI models to generate step-by-step explanations and math clues. AI can make mistakes.

Chemistry Question 126 – AP-EAMCET 2026

Clue 1 of 3: Root Mean Square Velocity and Temperature

🧠 Full Solution Path