The mean kinetic energy of one mole or gas per degree of

Options

A. `(1)/(2)kT`
B. `(3)/(2)kT`
C. `(3)/(2)kT`
D. `(1)/(2)RT`

Answer

To solve the question regarding the mean kinetic energy of one mole of gas per degree, we will follow these steps:

### Step 1: Understand the Mean Kinetic Energy Formula
The mean kinetic energy (\( U_{\text{average}} \)) of one mole of gas is given by the formula:
\[ U_{\text{average}} = \frac{1}{2} k N_A T \]
where:
- \( k \) is the Boltzmann constant,
- \( N_A \) is Avogadro's number,
- \( T \) is the absolute temperature in Kelvin.

### Step 2: Relate Boltzmann Constant to Gas Constant
We know that the Boltzmann constant (\( k \)) can be expressed in terms of the universal gas constant (\( R \)):
\[ k = \frac{R}{N_A} \]
where \( R \) is the gas constant.

### Step 3: Substitute \( k \) in the Mean Kinetic Energy Formula
Now, we can substitute the expression for \( k \) into the mean kinetic energy formula:
\[ U_{\text{average}} = \frac{1}{2} \left(\frac{R}{N_A}\right) N_A T \]
Here, \( N_A \) cancels out:
\[ U_{\text{average}} = \frac{1}{2} R T \]

### Step 4: Final Expression for One Mole of Gas
Since we are calculating the mean kinetic energy per mole, we can express it as:
\[ U_{\text{average}} = \frac{1}{2} R T \]

### Step 5: Identify the Correct Option
Now, we need to check the options provided:
1. \( \frac{1}{2} k T \)
2. \( \frac{3}{2} k T \)
3. \( \frac{3}{2} k T \)
4. \( \frac{1}{2} R T \)

From our calculation, we see that the correct answer is:
\[ \frac{1}{2} R T \]
This matches with option 4.

### Conclusion
Thus, the mean kinetic energy of one mole of gas per degree is:
\[ \frac{1}{2} R T \]

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