What is the ratio of kinetic energies of 3 g of hydrogen and 4 g of oxygen at T (K) ?

Options

A. `12 : 1`
B. `6 : 1`
C. `1 : 6`
D. `24 : 1`

Explanation

To find the ratio of kinetic energies of 3 g of hydrogen and 4 g of oxygen at temperature T (K), 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: Calculate the number of moles of hydrogen (H₂)
To find the number of moles of hydrogen, we use the formula:
\[ n = \frac{\text{mass}}{\text{molar mass}} \]
For hydrogen (H₂):
- Given mass = 3 g
- Molar mass of H₂ = 2 g/mol

Calculating moles of H₂:
\[ n_{H_2} = \frac{3 \text{ g}}{2 \text{ g/mol}} = 1.5 \text{ mol} \]

### Step 3: Calculate the number of moles of oxygen (O₂)
For oxygen (O₂):
- Given mass = 4 g
- Molar mass of O₂ = 32 g/mol

Calculating moles of O₂:
\[ n_{O_2} = \frac{4 \text{ g}}{32 \text{ g/mol}} = 0.125 \text{ mol} \]

### Step 4: Set up the ratio of kinetic energies
The ratio of the kinetic energies of hydrogen to oxygen can be expressed as:
\[ \frac{KE_{H_2}}{KE_{O_2}} = \frac{n_{H_2}}{n_{O_2}} \]
Substituting the values we calculated:
\[ \frac{KE_{H_2}}{KE_{O_2}} = \frac{1.5}{0.125} \]

### Step 5: Calculate the ratio
Now, we can simplify the ratio:
\[ \frac{1.5}{0.125} = \frac{1.5 \times 8}{1} = 12 \]

### Final Answer
Thus, the ratio of the kinetic energies of 3 g of hydrogen to 4 g of oxygen at temperature T (K) is:
\[ \text{Ratio} = 12 : 1 \]

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