How many moles of helium at temperature 300K and 1.00 atm pressure are needed to make the internal energy of the gas 100J?

Explanation

To find the number of moles of helium gas needed to achieve an internal energy of 100 Joules at a temperature of 300 K and a pressure of 1.00 atm, we can use the formula for the internal energy of a monatomic ideal gas.

### Step-by-Step Solution:

1. **Identify the properties of helium**:
Helium is a monatomic gas, which means it has a degree of freedom (f) of 3.

2. **Use the formula for internal energy**:
The internal energy (U) of a monatomic ideal gas can be expressed as:
\[ U = \frac{3}{2} n R T \]
where:
- \( U \) is the internal energy,
- \( n \) is the number of moles,
- \( R \) is the universal gas constant (approximately \( 8.31 \, \text{J/(mol K)} \)),
- \( T \) is the temperature in Kelvin.

3. **Rearrange the formula to solve for n**:
To find the number of moles (n), we rearrange the formula:
\[ n = \frac{2U}{3RT} \]

4. **Substitute the known values**:
Here, we know:
- \( U = 100 \, \text{J} \)
- \( R = 8.31 \, \text{J/(mol K)} \)
- \( T = 300 \, \text{K} \)

Plugging in these values:
\[ n = \frac{2 \times 100}{3 \times 8.31 \times 300} \]

5. **Calculate the denominator**:
First, calculate \( 3 \times 8.31 \times 300 \):
\[ 3 \times 8.31 = 24.93 \]
\[ 24.93 \times 300 = 7479 \]

6. **Calculate n**:
Now substitute back into the equation:
\[ n = \frac{200}{7479} \approx 0.0267 \, \text{moles} \]

7. **Final result**:
The number of moles of helium needed is approximately:
\[ n \approx 2.67 \times 10^{-2} \, \text{moles} \]

### Summary:
The number of moles of helium required to achieve an internal energy of 100 Joules at 300 K and 1.00 atm pressure is approximately \( 2.67 \times 10^{-2} \) moles.