At certain pressure and `127^(@)C` temperature the mean kinetic energy of hydrogen molecules is `8 xx 10^(-21) J`. What is the mean kinetic energy of hydrogen molecules at the same pressure and temperature `27^(@)C` ?

Explanation

To find the mean kinetic energy of hydrogen molecules at a temperature of 27°C, given that the mean kinetic energy at 127°C is \(8 \times 10^{-21} \, J\), we can follow these steps:

### Step 1: Understand the relationship between kinetic energy and temperature
The mean kinetic energy \(E\) of gas molecules is directly proportional to the absolute temperature \(T\) (in Kelvin). This relationship can be expressed as:
\[ E \propto T \]
This means that if we know the kinetic energy at one temperature, we can find it at another temperature using their respective absolute temperatures.

### Step 2: Convert temperatures from Celsius to Kelvin
To use the formula, we need to convert the given temperatures from Celsius to Kelvin:
- For 127°C:
\[ T_1 = 127 + 273 = 400 \, K \]
- For 27°C:
\[ T_2 = 27 + 273 = 300 \, K \]

### Step 3: Set up the proportionality equation
Using the proportionality established in Step 1, we can write:
\[ \frac{E_1}{E_2} = \frac{T_1}{T_2} \]
Where:
- \(E_1 = 8 \times 10^{-21} \, J\) (mean kinetic energy at 127°C)
- \(E_2\) is the mean kinetic energy at 27°C
- \(T_1 = 400 \, K\)
- \(T_2 = 300 \, K\)

### Step 4: Substitute the known values into the equation
Substituting the known values into the equation gives:
\[ \frac{8 \times 10^{-21}}{E_2} = \frac{400}{300} \]

### Step 5: Simplify the equation
Simplifying the right side:
\[ \frac{400}{300} = \frac{4}{3} \]
Thus, we have:
\[ \frac{8 \times 10^{-21}}{E_2} = \frac{4}{3} \]

### Step 6: Solve for \(E_2\)
Cross-multiplying to solve for \(E_2\):
\[ 8 \times 10^{-21} \cdot 3 = 4 \cdot E_2 \]
\[ 24 \times 10^{-21} = 4 \cdot E_2 \]
Now, divide both sides by 4:
\[ E_2 = \frac{24 \times 10^{-21}}{4} = 6 \times 10^{-21} \, J \]

### Final Answer
The mean kinetic energy of hydrogen molecules at 27°C is:
\[ E_2 = 6 \times 10^{-21} \, J \]

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