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 solve the problem of how many moles of helium at a temperature of 300 K and 1.00 atm pressure are needed to make the internal energy of the gas equal to 100 J, we can follow these steps:

### Step 1: Understand the relationship for internal energy
The internal energy \( U \) of an ideal gas is given by the formula:
\[ U = n \cdot C_v \cdot T \]
where:
- \( U \) is the internal energy,
- \( n \) is the number of moles,
- \( C_v \) is the molar heat capacity at constant volume,
- \( T \) is the temperature in Kelvin.

### Step 2: Identify the values given in the problem
From the problem, we have:
- \( U = 100 \, \text{J} \)
- \( T = 300 \, \text{K} \)

### Step 3: Determine \( C_v \) for helium
Since helium is a monatomic gas, the molar heat capacity at constant volume \( C_v \) is given by:
\[ C_v = \frac{3}{2} R \]
where \( R \) is the universal gas constant, approximately \( 8.314 \, \text{J/(mol·K)} \).

### Step 4: Substitute the known values into the internal energy equation
Substituting the values into the internal energy equation:
\[ 100 = n \cdot \left(\frac{3}{2} R\right) \cdot 300 \]

### Step 5: Solve for \( n \)
Rearranging the equation to solve for \( n \):
\[ n = \frac{100}{\left(\frac{3}{2} R\right) \cdot 300} \]
\[ n = \frac{100}{\frac{3}{2} \cdot 8.314 \cdot 300} \]
Calculating \( \frac{3}{2} \cdot 8.314 \cdot 300 \):
\[ = \frac{3 \cdot 8.314 \cdot 300}{2} = \frac{24942.0}{2} = 12471.0 \]
Now substituting back for \( n \):
\[ n = \frac{100}{12471.0} \approx 0.00802 \, \text{moles} \]

### Final Answer
Thus, the number of moles of helium needed is approximately:
\[ n \approx 0.00802 \, \text{moles} \]