Assume that helium behaves as an ideal monatomic gas. If 2 moles of helium undergo a temperature increase of 100 K at constant pressure, how much energy has been transferred to the helium as heat?

Options

A. `1700 J`
B. `2500 J`
C. `4200J`
D. `5000J`

Answer

To solve the problem step by step, we will use the formula for heat transfer at constant pressure for an ideal gas.

### Step 1: Identify the formula for heat transfer at constant pressure
The heat transferred to the gas at constant pressure (Qp) can be calculated using the formula:
\[ Q_p = n C_p \Delta T \]
where:
- \( n \) = number of moles of the gas
- \( C_p \) = molar heat capacity at constant pressure
- \( \Delta T \) = change in temperature

### Step 2: Determine the number of moles and change in temperature
From the problem, we know:
- \( n = 2 \) moles (given)
- \( \Delta T = 100 \) K (given)

### Step 3: Calculate the molar heat capacity \( C_p \) for a monatomic ideal gas
For a monatomic ideal gas, the molar heat capacity at constant pressure \( C_p \) is given by:
\[ C_p = \frac{5}{2} R \]
where \( R \) is the universal gas constant, approximately \( 8.314 \, \text{J/(mol K)} \).

### Step 4: Substitute values into the formula
Now we can substitute the values into the heat transfer formula:
\[ Q_p = n C_p \Delta T \]
\[ Q_p = 2 \left(\frac{5}{2} R\right) (100) \]

### Step 5: Simplify the equation
Substituting \( C_p \):
\[ Q_p = 2 \left(\frac{5}{2} \times 8.314\right) (100) \]
\[ Q_p = 2 \times 5 \times 8.314 \times 100 / 2 \]
\[ Q_p = 5 \times 8.314 \times 100 \]

### Step 6: Calculate the final value
Now calculate:
\[ Q_p = 5 \times 8.314 \times 100 \]
\[ Q_p = 4157 \, \text{J} \]

### Final Answer
The energy transferred to the helium as heat is approximately \( 4157 \, \text{J} \).

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