Question

The gases carbon monoxide (CO) and nitrogen at the same temperature have kinetic energies $ E_1 $ and $ E_2 $, respectively. Then,

• A. \( E_1 = E_2 \)
• B. \( E_1>E_2 \)
• C. \( E_1<E_2 \)
• D. None of these

Hint:

At the same temperature, all ideal gases have the same average kinetic energy. The kinetic energy is independent of the type of gas.

Answer 1

The Correct Option is A

To compare the kinetic energies \( E_1 \) and \( E_2 \) of carbon monoxide (CO) and nitrogen (N\(_2\)) at the same temperature, we must understand the relation between temperature and kinetic energy for gases.

According to the kinetic theory of gases, the average kinetic energy of an ideal gas's molecules is proportional to the absolute temperature of the gas. Mathematically, it is expressed as:

\[\bar{E} = \frac{3}{2}kT\]

where \(\bar{E}\) is the average kinetic energy per molecule, \(k\) is the Boltzmann constant, and \(T\) is the absolute temperature.

Since the problem states that both gases are at the same temperature, we have:

\[\frac{3}{2}kT_{\text{CO}} = \frac{3}{2}kT_{\text{N}_2}\]

Here, \(T_{\text{CO}} = T_{\text{N}_2} = T\), leading to the conclusion that:

\(E_1 = E_2\)

This indicates that the average kinetic energy per molecule is the same for both gases at the same temperature. 

Hence, the correct answer is: \( E_1 = E_2 \)

Answer 2

To solve the problem of determining the relationship between the kinetic energies of carbon monoxide (CO) and nitrogen gas (N₂) at the same temperature, we employ the kinetic theory of gases. According to this theory, the kinetic energy of a gas per mole at a given temperature is given by:

\( E = \frac{3}{2}kT \)

where \( k \) is the Boltzmann constant and \( T \) is the temperature in Kelvin. Under the assumption of identical conditions (same temperature), the expression for the kinetic energy does not depend on the mass or nature of the gas. For individual gas molecules, the average kinetic energy is given by:

\( E = \frac{3}{2} \cdot \frac{R}{N_A} \cdot T \)

Where \( R \) is the universal gas constant, and \( N_A \) is Avogadro's number. Since both gases are at the same temperature, the energies will be:

\( E_1 = \frac{3}{2}kT \) for CO

\( E_2 = \frac{3}{2}kT \) for N₂

This leads to the conclusion:

\( E_1 = E_2 \)

Thus, the option \( E_1 = E_2 \) is correct