Question

At T(K), the $v_{\text{rms}$ of CO$_2$ is 412 m/s$^{-1}$. What is its kinetic energy (in kJ mol$^{-1}$) at the same temperature? (CO$_2$ = 44 u)

• A. 3.7343
• B. 7.4687
• C. 14.9374
• D. 2.7343

Hint:

The kinetic energy of a gas can be calculated directly from $v_{\text{rms}}$ using $KE = \frac{1}{2} M v_{\text{rms}}^2$, where $M$ must be in kg/mol for consistency with SI units.

Answer 1

The Correct Option is A

Let’s break this down step by step to calculate the kinetic energy of CO$_2$ and determine why option (1) is the correct answer.
Step 1: Understand the relationship between $v_{\text{rms}$ and kinetic energy}
The kinetic energy per mole of an ideal gas is:
\[ KE = \frac{1}{2} M v_{\text{rms}}^2 \]
where $M$ is the molar mass in kg/mol, and $KE$ is in J/mol.
Step 2: Identify the given values and calculate the kinetic energy

• $v_{\text{rms}} = 412 \, \text{m/s}$
• Molar mass of CO$_2$, $M = 44 \, \text{g/mol} = 0.044 \, \text{kg/mol}$

\[ KE = \frac{1}{2} \times 0.044 \times (412)^2 \]
\[ (412)^2 = 169744 \]
\[ KE = 0.022 \times 169744 = 3734.368 \, \text{J/mol} \]
Convert to kJ/mol:
\[ KE = \frac{3734.368}{1000} = 3.734368 \, \text{kJ/mol} \]
Step 3: Confirm the correct answer
The calculated kinetic energy is 3.7343 kJ/mol, which matches option (1).
Thus, the correct answer is (1) 3.7343.