The ratio of the kinetic energies of equal number of moles of `H_(2)` and He at the same temperature is

Options

A. `1:2`
B. `2:1`
C. `1:1`
D. `4:1`

Answer

To solve the problem of finding the ratio of the kinetic energies of equal number of moles of \( H_2 \) (hydrogen) and He (helium) at the same temperature, we can follow these steps:

### Step 1: Understand the formula for kinetic energy
The kinetic energy (KE) of a gas can be expressed using the formula:
\[ KE = \frac{3}{2} nRT \]
where:
- \( n \) = number of moles of the gas
- \( R \) = universal gas constant
- \( T \) = absolute temperature in Kelvin

### Step 2: Write the kinetic energy expressions for both gases
For hydrogen (\( H_2 \)):
\[ KE_{H_2} = \frac{3}{2} n_{H_2} RT \]
For helium (He):
\[ KE_{He} = \frac{3}{2} n_{He} RT \]

### Step 3: Set the number of moles equal
Since the question states that we have equal numbers of moles of \( H_2 \) and He, we can denote them as \( n \):
\[ n_{H_2} = n_{He} = n \]

### Step 4: Substitute into the kinetic energy formulas
Now we can substitute \( n \) into the kinetic energy formulas:
\[ KE_{H_2} = \frac{3}{2} nRT \]
\[ KE_{He} = \frac{3}{2} nRT \]

### Step 5: Calculate the ratio of the kinetic energies
Now, we can find the ratio of the kinetic energies of hydrogen to helium:
\[ \text{Ratio} = \frac{KE_{H_2}}{KE_{He}} = \frac{\frac{3}{2} nRT}{\frac{3}{2} nRT} = 1 \]

### Conclusion
Thus, the ratio of the kinetic energies of equal number of moles of \( H_2 \) and He at the same temperature is:
\[ \text{Ratio} = 1:1 \]

### Final Answer
The ratio of the kinetic energies of equal number of moles of \( H_2 \) and He at the same temperature is \( 1:1 \).

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