At what temperature will the molar kinetic energy of `0.3 mol` of `'He'` be the same as that of `0.4 mol` of argon at `400 K`?

Options

A. (a) 700 K
B. (b) 533 K
C. (c) 800 K
D. (d) 400 K

Explanation

To solve the problem, we need to find the temperature at which the molar kinetic energy of 0.3 moles of helium (He) is equal to that of 0.4 moles of argon (Ar) at 400 K.

### Step-by-Step Solution:

1. **Understand the Molar Kinetic Energy Formula**:
The molar kinetic energy (KE) of a gas can be expressed as:
\[ KE = \frac{3}{2} nRT \]
where \( n \) is the number of moles, \( R \) is the universal gas constant, and \( T \) is the temperature in Kelvin.

2. **Set Up the Equation**:
We know that the molar kinetic energy of helium must equal that of argon. Therefore, we can write:
\[ \frac{3}{2} n_{He} R T_{He} = \frac{3}{2} n_{Ar} R T_{Ar} \]
Here, \( n_{He} = 0.3 \) moles, \( n_{Ar} = 0.4 \) moles, and \( T_{Ar} = 400 \) K.

3. **Cancel Common Terms**:
The constants \( \frac{3}{2} \) and \( R \) are common on both sides of the equation and can be canceled out:
\[ n_{He} T_{He} = n_{Ar} T_{Ar} \]

4. **Substitute Known Values**:
Substitute the values of \( n_{He} \), \( n_{Ar} \), and \( T_{Ar} \) into the equation:
\[ 0.3 T_{He} = 0.4 \times 400 \]

5. **Calculate the Right Side**:
Calculate \( 0.4 \times 400 \):
\[ 0.4 \times 400 = 160 \]

6. **Solve for \( T_{He} \)**:
Now, substitute back into the equation:
\[ 0.3 T_{He} = 160 \]
To find \( T_{He} \), divide both sides by 0.3:
\[ T_{He} = \frac{160}{0.3} = 533.33 \, K \]

### Final Answer:
The temperature at which the molar kinetic energy of helium is equal to that of argon is approximately **533.33 K**.

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