At what temperature will the total kinetic energy of 0.5 moles of He be the same as the total kinetic energy of 0.6 moles of neon at 300K?

Options

A. 300 K
B. 360 K
C. 400 K
D. 250 K

Explanation

To solve the problem, we need to find the temperature at which the total kinetic energy of 0.5 moles of helium (He) is equal to the total kinetic energy of 0.6 moles of neon (Ne) at 300 K.

### Step-by-Step Solution:

1. **Understand the Kinetic Energy Formula**:
The kinetic energy (KE) of an ideal gas can be expressed by the formula:
\[ KE = \frac{3}{2} nRT \]
where:
- \( n \) = number of moles
- \( R \) = universal gas constant
- \( T \) = temperature in Kelvin

2. **Set Up the Equation**:
According to the problem, the kinetic energy of helium (He) must equal the kinetic energy of neon (Ne):
\[ KE_{He} = KE_{Ne} \]
Therefore, we can write:
\[ \frac{3}{2} n_{He} R T_{He} = \frac{3}{2} n_{Ne} R T_{Ne} \]
Since \(\frac{3}{2} R\) is a common factor, we can simplify the equation to:
\[ n_{He} T_{He} = n_{Ne} T_{Ne} \]

3. **Insert Known Values**:
We know:
- \( n_{He} = 0.5 \) moles
- \( n_{Ne} = 0.6 \) moles
- \( T_{Ne} = 300 \) K
We need to find \( T_{He} \):
\[ 0.5 \cdot T_{He} = 0.6 \cdot 300 \]

4. **Calculate \( T_{He} \)**:
Rearranging the equation gives:
\[ T_{He} = \frac{0.6 \cdot 300}{0.5} \]
Now, calculate:
\[ T_{He} = \frac{180}{0.5} = 360 \text{ K} \]

5. **Final Answer**:
The temperature at which the total kinetic energy of 0.5 moles of helium equals the total kinetic energy of 0.6 moles of neon at 300 K is:
\[ T_{He} = 360 \text{ K} \]