Two flasks A and B of equal volume containing equal masses of `H_2 and CH_4` gases are at 100 K and 200 K temperature, respectively. Which of the following is true about the total KE (kinetic energy) ?

Options

A. Total KE of `H_2` is four times that of `CH_4`
B. Total KE of `CH_4` is four times that of `H_2`
C. Total KE of `H_2` is two times that of `CH_4`
D. Total KE of `CH_4` is two times that of `H_2`

Answer

To solve the problem regarding the total kinetic energy (KE) of the gases in flasks A and B, we will follow these steps:

### Step 1: Identify the given data
- Flask A contains Hydrogen gas (H₂) at a temperature of 100 K.
- Flask B contains Methane gas (CH₄) at a temperature of 200 K.
- Both flasks have equal volumes and contain equal masses of gases.

### Step 2: Calculate the number of moles of each gas
- The molar mass of H₂ = 2 g/mol.
- The molar mass of CH₄ = 16 g/mol.
- Let the mass of each gas in the flasks be 'm'.

Using the formula for moles:
\[ n = \frac{m}{M} \]
- Moles of H₂, \( n_{H2} = \frac{m}{2} \)
- Moles of CH₄, \( n_{CH4} = \frac{m}{16} \)

### Step 3: Write the formula for total kinetic energy
The total kinetic energy (KE) of an ideal gas can be expressed as:
\[ KE = \frac{3}{2} nRT \]
Where:
- \( n \) = number of moles
- \( R \) = universal gas constant
- \( T \) = temperature in Kelvin

### Step 4: Calculate the total kinetic energy for each gas
- For H₂ at 100 K:
\[ KE_{H2} = \frac{3}{2} \left(\frac{m}{2}\right) R (100) \]
- For CH₄ at 200 K:
\[ KE_{CH4} = \frac{3}{2} \left(\frac{m}{16}\right) R (200) \]

### Step 5: Simplify the expressions for kinetic energy
- For H₂:
\[ KE_{H2} = \frac{3mR}{4} \times 100 = \frac{300mR}{4} = 75mR \]
- For CH₄:
\[ KE_{CH4} = \frac{3mR}{32} \times 200 = \frac{600mR}{32} = \frac{75mR}{4} \]

### Step 6: Compare the total kinetic energies
Now, we can compare the two kinetic energies:
- \( KE_{H2} = 75mR \)
- \( KE_{CH4} = \frac{75mR}{4} \)

### Step 7: Determine the relationship
From the calculations:
\[ KE_{H2} = 4 \times KE_{CH4} \]
This means the total kinetic energy of H₂ is four times that of CH₄.

### Conclusion
The total kinetic energy of the Hydrogen gas (H₂) is greater than that of Methane gas (CH₄) by a factor of 4.

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